What is a manifold? For complete dummies when it comes to space-time, what is a manifold and how can space-time be modelled using these concepts?
 A: The common definition of an $n$-manifold is: a topological space that resembles Euclidean space in a neighborhood of each point (and a manifold is any $n$-manifold). This means that if you take an arbitrary point in the manifold, there always exists a small enough ball around the point within which the space can be continuously deformed into flat space. A simple example is a circle. This is a $1$-manifold because if you take any connected subspace of it, you can straighten it out to a line (Euclidean $1$-space) even though you can't do this with the entire circle. The same could be said of any simple smooth curve ("simple" means non-self-intersecting and "smooth" means differentiable). Similarly, the sphere, torus, and other smooth surfaces are $2$-manifolds.
A more subtle point is the distinction between a manifold and its embedding. A powerful theorem of Nash shows that for every $n$-manifold $M$, there exists some $m\geq n$ such that $M$ embeds into $\mathbb{R}^m$. In the abstract sense, one could parametrize a curve in a way that satisfies the definition of a manifold even if all its embeddings into $\mathbb{R}^2$ are self-intersecting (maybe it embeds without intersection in $\mathbb{R}^3$). But a fixed embedding of a self-intersecting curve is technically not a manifold because it has a point that look like a $+$ (not $\mathbb{R}$). Similarly, the Klein bottle can be embedded in $\mathbb{R}^4$ as a $2$-manifold even though its embeddings into $\mathbb{R}^3$ are self-intersecting.
Spacetime is $4$-dimensional. The usual application of the manifold concept fits in here if we think of the spacial part of spacetime deforming continuously over time. In this model, at any fixed moment in time the universe can be thought of as a $3$-manifold, but it must satisfy additional constraints. For one thing, we assume the universe to be homeomorphic and isotropic. For another, such a $3$-manifold must make sense as a cross-section of the $4$-dimensional structure.
The $3$-dimensional homeomorphic and isotropic manifolds have been a very active subject of mathematical research, due to the seminal work of Bill Thurston starting in the 1970's. Among these manifolds, there is one that is flat (Euclidean $3$-space), one with positive curvature (the $3$-sphere), and there are infinitely many hyperbolic structures. Some mathematicians believe that the spacial portion of spacetime can be modeled with a hyperbolic manifold, though this is not widely believed in physics (explained below). In the early 80's, Jeff Weeks discovered the closed hyperbolic $3$-manifold of minimal volume, and it was hoped by some that this was a model for the universe, however it failed to satisfy the requirements for being a spacial cross-section of spacetime. More recently, based on the data on microwave background radiation, Weeks conjectured that the correct model is Poincare dodecahedral space (like a $12$-sided die where every time you leave through one face, you come back in through another with some rotation), which is also hyperbolic. 
Many physicists believe the universe to be flat based on our measurements of the curvature of the observable portion of it. However (and this statement is biased, coming from my perspective as a mathematician specialized in topology), if the observable universe is a relatively tiny portion of the general universe, then the definition of a manifold tells us that we should expect it to look flat regardless of its actual topological structure. It remains a topic of interest what the topology is of (the spacial portion of) the general universe in spacetime, as a manifold.
Alternatively, one could consider the universe, with time, as a $4$-manifold, though these are not as well understood. There are also higher-dimensional theories of the universe in physics. In mathematics there is no limitation on $n\in\mathbb{N}$ in defining an $n$-manifold. There are also well-developed theories of infinite-dimensional manifolds (e.g. Banach manifolds), as well as $n\in\mathbb{Q}^+$, i.e. fractional-dimensional manifolds (fractals), but these concepts are less related to the spacetime model.
A: There's been very good answers, and they've depicted very well, and conceptually as well as accurately, what a manifold is, how it can be used to describe inherently curved space, and how the idea of continuity and differentiability arise by joining together all the local charts. 
I would like to depict the way in which physicists got to 4 dimensional spacetime. Including time was critical, and was the jumping off point from special relativity.  Or to say it differently, why it made sense physically to use a geometrical construct, and Riemannian geometry in particular, to model/describe a dynamic space and time, spacetime, in a manner consistent with special relativity and with gravity. I may not be totally accurate in the history, but it is closer to the physical thinking. And it explains more physically  then how manifolds, or Riemannian geometry, is an appropriate construct to describe spacetime and it's dynamics. It was not simply that it was a Riemannian entity, but that it was a live dynamic entity.   
Historically this arose a little differently in physics than in math, although basically meaning the same things, and with the math then developed for Riemannian geometry a key tool to get to the equations and think through the concepts. In physics the idea of spacetime first arose with Einstein, and it was flat spacetime (although the name spacetime came later), with a Lorentzian (and flat) metric which was the appropriate one to use in special relativity. There was initially no thought that the spacetime, or rather the compilation of x, y, z and t coordinates in any way defined a dynamic or varying entity. It was simply thought that any inertial observer could pick theirs, and we had the Lorentz transformations. 
Einstein (with others possibly contributing, there is some discussion on how much), through a history that is described in various books, came to the equivalence principle that a gravitational force seemed to have the same effect as simply being in a constantly accelerated reference frame, and Einstein was led to then look at (pseudo) Riemannian geometry as a way of describing the motion in terms of geodesics, and what became known as spacetime as a 4 dimensional pseudo-Riemannian entity, with the physics independent of the coordinate systems chosen, i.e., of the observer. And by seeing that in the weak field limit and low velocities it reduced to Newtonian gravity, and some other things, he got the field equations. He did use the full construct of Riemannian geometry, which was not well known at that point but was there. He got to it because he needed to describe something independently of coordinate systems, whether inertial, accelerating or anything else. I don't know if the word manifold for it existed then. But differentiability was part of a Riemannian 'space'. 
That geometry is able to describe gravity is a pretty deep finding by Einstein and some of his colleagues. It had to do with the equivalence principle, which comes essentially from the equivalence of inertial (motion, change in spacetime) and gravitational mass (force). That equivalence does not hold for any of the other forces and so far it's been impossible to unify Einstein's gravitational theory with any of the other forces. 
A: This is my non-physicists take. A manifold is a curved space that is locally flat. Think of the surface of the Earth, which is a two-dimensional manifold (can be described using two coordinates - latitude and longitude). Small patches of the Earth's surface can be described using Euclidean geometry; bigger areas can't as this geometry breaks down.
In the context of relativity, the manifold (a) has four dimension (three of space and one of time) and is called spacetime; (b) is differentiable; and (c) is described by a function called a metric which gives the time difference and distance between infinitesimally close points. Different coordinate systems have different metrics describing the same distance between infinitesimally close points. Using the metric, you can construct a four-index tensor called the Riemann curvature tensor. If and only if that tensor is zero is the space at that point flat, otherwise it's curved.
Special relativity deals with flat spacetime (called Minkowski spacetime), i.e. deals with situations where the effects of gravity are negligible. Mass-energy curves spacetime. Free bodies or light rays will follow the shortest path (aka geodesic) between two points in spacetime. In order to calculate that path you need to know the metric. The two most common metrics are the Minkowski metric (describing flat spacetime) and the Schwarzschild metric (describing the spacetime around a spherically symmetric object such as our Sun).
A: To introduce the concept of a smooth manifold, I will first introduce topological manifolds.

Topological Manifold
We say that $M$, a topological space, is also a topological manifold if,


*

*For any two points I pick, say $p,q \in M$, there are disjoint open subsets $U$ and $V$ of the space $M$ such that $p \in U$ and $q\in V$. In other words they can be separated by neighbourhoods.

*There exists a countable basis for the topology of $M$, which is to say that we can construct any open set in $M$ from the union of a bunch of other open sets, called the basis $B$, and that this basis is countable.

*Crucially, every point $p \in M$ has a neighbourhood that is homeomorphic to an open subset of $\mathbb{R}^n$. In other words there exists a continuous function with continuous inverse from that neighbourhood to an open set in $\mathbb R^n$ and this is what we mean by locally Euclidean.


To stress again, we can pick any $p\in M$ and an open set $U \subset M$ containing $p$, and we are guaranteed to be able to construct a homeomorphism $\psi : U \to \tilde U$ where $\tilde U \subset \mathbb R^n$.  In addition, this definition of locally Euclidean is totally equivalent to being able to construct a homeomorphism to an open ball in $\mathbb R^n$ or $\mathbb R^n$ itself. The first two requirements are rather formal, and for what follows the third is crucial to understand.

Charts
To proceed with constructing the notion of a smooth manifold, we introduce coordinate charts. In particular, a coordinate chart is a pair $(U, \varphi)$ where $U \subset M$ is an open set and $\varphi(U) \subset \mathbb R^n$ is the homeomorphism we spoke of, to $\mathbb R^n$.
The map $\varphi$ is a local coordinate map whose components are coordinates and $U$ is the coordinate neighbourhood.

Smooth Structure
To be able to do calculus on such a manifold, we have to add a smooth structure to it. If $(U,\varphi)$ and $(V, \psi)$ are two charts such that $U \cap V \neq \varnothing$, then the map
$$\psi \circ \varphi^{-1} : \varphi(U \cap V) \to \psi(U \cap V),$$
called the transition map, is a homeomorphism. The two charts are smoothly compatible if this transition map is a diffeomorphism, which is to say all components have partial derivatives to all orders, is bijective, and the inverse is continuous.
We can define an atlas $\mathcal A$ as the collection of charts which cover the entire manifold, so that any point must belong to the domain of one of these charts. Notice this means we do not require that one coordinate system cover the entire manifold.
You may guess now we will call $\mathcal A$ a smooth atlas if all charts are smoothly compatible as defined above.
Before we get to the punchline, we could have a manifold $M$ that has many smooth atlases, so in the following definition, we choose the maximal one or the one that is complete in the sense that every chart that is smoothly compatible is contained in $\mathcal A$.
A smooth manifold is thus the pair $(M, \mathcal A)$, and we can define a function $f : M \to \mathbb R^n$ to be smooth if $f \circ \varphi^{-1}$ is smooth for each chart.

How do we model space-time using manifolds?
In the theory of general relativity, we treat space-time as a Riemannian manifold which gives further constraints on the notion of a smooth manifold.
Each Riemannian manifold comes equipped with the metric tensor $g_p(X,Y)$ which takes two tangent vectors $X,Y \in T_p M$ which lie in the tangent space at the point $p$, and it gives us the notion of the length of a vector and the angle between vectors in a generalised way.
The Einstein field equations which relate matter to the curvature of space-time modelled as a manifold, explicitly depend on this metric $g$.
A: What is a manifold?
A manifold is a concept from mathematics that has nothing to do with physics a priori. 
The idea is the following: You have probably studied Euclidean geometry in school, so you know how to draw triangles, etc. on a flat piece of paper. In contrast to common parlance, let's take "space" to mean anything with a number of points. The Euclidean plane ($\mathbb{R}^2$) or your piece of paper are a "space", the 3d-space around you is a "space" or the surface of the world is a "space" (caveat: Actually, I want to define a topological space, which is not "everything with a number of points", but let's not get distracted here). 
Now, if you look at the surface of the sphere, it's definitely not a Euclidean space: In Euclidean geometry, the sum of every angle in a triangle is 180° which is not true for the surface of a ball, a sphere. However, if you only look at a small patch of the sphere, it is approximately true. For instance, you perceive the earth as flat although it isn't if you look from above. 
A manifold is every "space" with this property: locally, it looks like a Euclidean plane. The circle is a manifold (it looks like a line locally, which is the one-dimensional Euclidean space $\mathbb{R}$), the sphere (it looks like a plane locally), your room (it looks like a 3d-Euclidean space $\mathbb{R}^3$ locally - forget about the boundaries here), etc. 
The cool thing about manifolds is that this property of looking like Euclidean space locally makes it possible to describe them completely using only Euclidean spaces. Since we know Euclidean space very well, that's a good thing. For instance, you can take a map of England - since the word "map" is used differently in mathematics, let's call it a "chart". This is a perfectly good way of describing England, although it really is part of a round object. You can patch a lot of these charts together to get a whole atlas covering the earth which gives you a nice description of the earth using only 2d pieces of paper. Obviously, you'll need more than one chart to cover the whole earth without doubling certain points and obviously, if the chart covers a very large area, it will look very distorted at some places, but it's possible as you can see.
And that's a manifold. It's some space where you can create an atlas of charts, each of which is a (part of a) Euclidean space describing a part of the space. Okay, not quite: what you want of the manifold is that you can get from chart to chart with a nice operation. For instance, in your atlas of the earth, some charts will overlap and points in the overlap that are close together on one chart will be close together on the other chart. In other words, you have a map between the overlapping regions of any two charts and that map is continuous (at that point you get a topological manifold) or even differentiable (at that point you get a differentiable manifold). 
By now, it should be obvious to you that it should be possible to say that the space around us is a differentiable manifold. It seems perfectly accurate to describe it using $\mathbb{R}^3$ locally, as you have probably done in school. And that's also how manifolds enter relativity: If you add the time dimension, it turns out to be a good guess, that you can still model the space + time as a four dimensional manifold (meaning every chart looks like $\mathbb{R}^4$ locally). 
Why model spacetime with manifolds?
Now you know what a manifold is, but even if you get an idea of how you could model spacetime as a manifold, this doesn't really tell you why you should model spacetime as a manifold. After all, just because you can do something, that doesn't always make it particularly useful.
Consider the following problem: Given two points, what's their shortest distance? 
[Aside: Before answering this question, I want to mention that although I talked about things like distances and angles before, you don't necessarily have these concepts on an arbitrary manifold because it might be impossible to define something like this for your underlying "space", but if you have a "differentiable manifold" (meaning that the functions that get you from chart to chart in the overlapping regions are differentiable), then you do. At that point, it becomes possible to speak about distances. For physics, especially general relativity, you always have a notion of distances and angles.]
Back to the problem of shortest distance: In $\mathbb{R}^n$, the answer is pretty simple. The smallest path between two lines is the straight line between them. But on a sphere? In order to define this, you first need a distance on the sphere. But how to do this? At that point I'd already know what the shortest distance is! 
Here is one idea: If you consider a flight from London to Buenos Aires (for example), what's the "shortest path"? Well, the earth is more or less a sphere in some $\mathbb{R}^3$. That's a Euclidean space, so you know how to compute distances there, so the shortest path is just the smallest distance of all possible paths. Easy. However, there is a problem: This only works because we have some ambient three dimensional space. But that doesn't have to be the case - indeed our own "space" doesn't seem to be embedded in some four spatial dimensions dimensional hyperspace (or whatever you want to call it). 
Here is another idea: Your manifold locally looks like a Euclidean space where the answer is simple. What if you only define your distance locally and then somehow patch it together so that it makes sense? 
The beautiful thing is that a differentiable manifold gives you tools to do that. This way, you can create a measure of distance (called a Riemannian metric), which allows you to calculate shortest paths between points even without ambient spaces. But it doesn't stop there. What are parallel lines? What happens to a local coordinate system? For instance, if you fly with your plane, it seems that you are always looking ahead, yet your field of view doesn't go in a straight line, how does your field of view change going along a path? Once you have your metric, it's all straightforward. 
It should be clear that all of these questions are questions that you can ask about the space(time) surrounding you - and you'd want the answer to them! It also seems natural that you should actually be able to answer these questions for our universe. 
So, what's the metric of our space? Can we just patch it together locally? Well, we could, but it's not going to be unique, so how to decide what is the right metric? That's exactly what general relativity is about: The fundamental equations of general relativity tell us how the distance measure in space time is related to matter and energy.
A little bit more about topology (in case you are interested)
Finally, if you want to learn more about the "space" aspect that I left out above, let's have a closer look there. What you want is not any set of points, but a set of points which has neighbourhoods for every point. You can think of a neighbourhood of a point as a number of points which are somehow "near" the point. Just like in real life, your neighbourhood could be really big, it could comprise all of the space, it mustn't even be connected, but it must somehow always comprise the points immediately "next to" you. In fact, if you have a distance measure such as the usual Euclidean distance in $\mathbb{R}^n$, then a set of neighbourhoods is given by all balls of all sizes around any point. However, you can define these neighbourhoods also without having a distance measure, but you can still somehow think of "nearness". 
These spaces are enough to let you define "continuous functions", where a function is continuous at a point, if all points "near" this point (meaning in some neighbourhood) stay "near" to the point after the mapping (meaning they are mapped into some neighbourhood again). Usually and especially for all manifolds we really want to talk about in relativity, you'd add some more conditions to the spaces to have nicer properties, but if you want to know about this, I suggest to begin learning the true mathematical definitions. There are a lot of other answers that cover the basics!
A: The manifold is a mathematical concept.

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an $n$-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension $n$.

Its use allows for generalizations from the classical Euclidean spaces, and as General Relativity by construction distorts space and time, a manifold  is the proper word to describe  the "coordinates" as distorted from euclidean spaces.
A: Historically, manifolds grew out of the following idea.
We often study various surfaces like the sphere or the cylinder by placing them in three dimensional Euclidean space, and from there study geometry. However, there are oddities:


*

*Curves don't really have any 'internal' shape, but there is all sorts of curvature that arise from how we draw the curve in Euclidean space!

*The usual cylinder is a flat surface despite the fact one might naively look at it and think its circular shape means that it's curved.

*We know from spherical geometry and hyperbolic geometry that shapes can have some nontrivial intrinsic geometry.


So there is a nontrivial problem here of distinguishing between what parts of geometry are actually intrinsic to the shape we study, and what parts of geometry are extrinsic — accidents of how we place the shape in Euclidean space.
The idea of a manifold was invented to solve this problem — it gives a useful way to work with interesting shapes in a purely intrinsic way, which ensures that all of the geometry we study in that fashion is truly intrinsic to the manifold.
The underlying idea is to cover the shape with coordinate charts, and describe the geometry using calculus on the coordinate charts. Think of using maps to represent the surface of the Earth. 

Now for physics, manifold come into play in the completely opposite fashion. We have centuries of experience doing physics on what is basically coordinate charts, and we know that's roughly how the universe looks on sufficiently small scales, but the large scale topology of the universe may well be more involved than that.
Enter manifolds, a pre-made mathematical theory about how to combine coordinate charts together to describe a more interesting topological space.

Even if one is not interested in more interesting manifolds, they still come into play due to modern mathematicsl — differential geometry is the language for doing sophisticated calculations in multivariable calculus, particularly when geometric ideas are involved, and the theory and practice of differential geometry is generally developed on manifolds.
A: There are already several good answers. So I will try to write a short answer that just answers the question without any detailed discussions.
After the invention of Special Relativity, Einstein tried to invent a Lorentz-invariant theory of gravity, but without any success. Ultimately, the problem was solved by replacing the Minkowski space-time with a curved space-time, i.e., by the geometrization of gravity. In a curved space-time the curvature is generated (and reacted back upon) by energy and momentum.
A manifold is one of the fundamental concepts in mathematics, in particular geometry. We are all aware of the n-dimensional Euclidean space $\mathbb{R}^n$ and the set of n-tuples $(x^1,x^2,...,x^n)$. The notion of a manifold captures the idea of a space which may be curved and can have a complicated topology, but which resembles the topology of $\mathbb{R}^n$ in local regions. The entire manifold is constructed by smoothly sewing together these local regions.
The manifold structure thus provides a natural setting upon which the theory of gravity can be built based on Einsteins Equivalence Principle: The curved space-time resembles a flat space-time locally where the laws of Special Relativity holds good.
A: A manifold is defined in several steps:

*

*It is a topological space that is Hausdorff and second countable. This sounds technical and it is. But what it means is that the space is continuous, that when we focus down to a point what we see is exactly one point and not several points close together (or even, far apart) and that we exclude certain very large spaces - for example, the long line which is much, much longer than our universe and any concievable universe.


*That is topologically speakimg,  locally Euclidean. This means when we focus down to small areas these look like affine space.
This is the definition of a topological manifold. Its worth pointing out that we can add manifolds together - this is the disjoint sum. So for example, adding two spheres together, we get exactly a space consisting of two spheres. We can also multiply manifolds: a lime multiplied by a circle gives a cylinder whilst a circle multiplied by another circle gives a torus. Hence we have a certain arithmetic of manifolds.
Usually in physics, especially after Newton's discovery of calculus, we ask for smoothness.
First we note that manifolds have an atlas of charts, this comes from the second condition. And hence we have chart changes where the charts overlap. These are called transition maps. These by definition are continuous as we defined everything in topological terms.


*A smooth manifold is a topological manifold where all the transition maps are smooth.

In a similar manner or by extra structure we can define many other manifolds such as complex manifolds, paracomplex manifolds, Riemannian manifolds etc. There's a whole bestiary of them ...
Spacetime is modelled as a Riemannian manifold. This simply means that locally we have a metric and so locally we can measure angles and distances. It was Minkowski who first suggested that spacetime was best thought of in this way after Einstein dreamt up special relativity. Although Einstein did not take to this at first, he relied on it to develop General Relativity.
Often this geometry is hidden in the usual physicists language. For example,  a contravariant vector at a point is defined by its transformation properties there. In the geometric language, this is exactly a tangent vector at that point of spacetime. This is an invariant notion.
In our own flat space, a parallelogram closes up. In curved space thos need no longer be the case and there is a defect that is measured by the curvature. We can turn this around and measure the curvature by how parallelograms don't (or do) close up.
Riemann actually speculated that the metric of space and time might vary on small scales. A picture that is elaborated in Wheelers spacetime foam. It was Clifford, who having read Riemanns thesis, speculated that all forces, that would be gravity and electromagnetism in his time, was reducible to changes in the metric and hence related to curvature. His bold hypothesis turned out to be completely correct as verified spectacularly by Einstein in the case of GR and less spectacularly by a whole host of theorists for electromagnetism and the weak and the strong force.
