You have many comments to the effect that "topology is needed to describe continuity, calculus concepts, the notion of "looks like", homeomorphism and so forth". And these are all altogether right, but I'm getting that your question is about the global picture. Also, the following is mainly about a toplological or differentiable manifold; Joshphysics's link shows that there are many other concepts of manifold.
We begin with the notion of "locally looks like $\mathbb{R}^N$"; but you can have a set $\mathbb{M}$ of any kind of weird creatures whose subsets you can put into one-to-one, surjective correspondence with some open (more about this below) subset of $\mathbb{R}$ (wontedly a simply connected neighbourhood of the origin). For one of these subsets $\mathcal{N}$ you have a "labeller" map $\lambda:\mathcal{N}\to\mathbb{R}^N$. Then you notions of open, neighbourhood and all the rest of it arise by definition: a subset $\mathcal{O}\subset\mathcal{N}$ is open iff $\lambda(\mathcal{O})$ is open in $\mathbb{R}^N$. Likewise a "path" $\sigma:\mathbb{R}\to\mathcal{N}$ is $C^0,\,C^1\, C^\omega$ or whatever iff $\lambda\circ \sigma:\mathbb{R}\to\mathbb{R}^N$ has the same property. All topology, neighbourhood, calculus, differentiability and so forth concepts are then defined by "fiat", and the need for the concepts is why we want our zoo of weird creatures to "locally look like $\mathbb{R}^N$" in the first place so this is all highly intuitive and obvious.
So I'm guessing (also by reading your other probing questions on this site) that you already understand all this. So the crucial question is then that of transistion maps and how we glue all our local copies of $\mathbb{R}^N$ together. Going back to our subset $\mathcal{N}\subset\mathbb{M}$:there are other "local copies" of $\mathbb{R}^N$ that bestow our topological / calculus and so forth concepts on subsets of $\mathbb{M}$ other than $\mathcal{N}$. But these subsets must overlap, because, when we're doing calculus or toplogy or dynamics or whatever, we don't want suddenly to run into a "co-ordinate wall" and have to jump suddenly from one co-ordinate system to another. As an example, suppose we have a spacecraft in an Einstein manifold (universe that is a vacuum solution to the EFEs). For calculus, measurement and other mathematical concepts, we need always to be able to define the manifold in a neighbourhood around the spaceship: so, as the spaceship nears the boundary of one co-ordinate system, it must also be describable by another co-ordinate system wherein we can carve out a "neighbourhood": we couldn't do this if our co-ordinate systems didn't overlap but instead partitioned the manifold $\mathbb{M}$. Otherwise put, in relativity, the boundary between co-ordinate systems is an artifact of our particular mathematical description of the physics, it does not belong to the physics. Another, dramatic, example is the phenomenon of gimbal lock in Euler angle charts for the unit sphere that very nighly cost the Apollo 11 astronauts their lives, cost many pilots their lives in the years before then and is the reason why the software processing signals from fibre ring Sagnac gyros that keep you safe in a commercial jetliner either manipulate the aeroplane's calculated orientation in two overlapping charts covering $SO(3)$ or, more recently, model the aeroplane's orientation by unit quaternions in $SO(3)$'s double cover $SU(2)$.
So, our overlap is very much needed, so many, if not all, regions in the manifold can be described by more than one local copy of $\mathbb{R}^N$ with more than one labeller. So, suppose we have two regions $\mathcal{N}_1,\,\mathcal{N}_2$ with labellers $\lambda_1:\mathcal{N}_1\to\mathbb{R}^N$, $\lambda_2:\mathcal{N}_1\to\mathbb{R}^N$: we must make sure that these labellers yield consistent notions of opennes, neighbourhood, differentiability and all the rest of it in a region $\mathcal{N}_1\cap\mathcal{N}_2$. So, a set $\mathcal{O}\subseteq\mathcal{N}_1\cap\mathcal{N}_2$ must be open as reckonned by labeller $\lambda_1$ and $\lambda_2$ and so $\lambda_1\circ\lambda_2^{-1}$ and $\lambda_2\circ\lambda_1^{-1}$, the "transition maps" between charts, must be local homeomorphisms, analytic, diffeomorphisms, or whatever the relevant notion is for the kind of manifold in question. Likewise for all other calculus and topological concepts we wish to speak of. This is most readily achieved if the charts (ranges of the labellers $\lambda_j$) are open, and their intersections are open as reckonned by all local copies of $\mathbb{R}^N$ that are applicable to the overlap. So we have two axioms for manifolds further to the obvious one that every point in the manifold must belong to the preimage of at least one labeller:
An intersection between two "patches" (domains of labellers) must be open in the topology as reckonned by each of the two labellers for the overlapping charts;
The transition maps must be local homeomorphisms, diffeomorphisms, ....
Some authors also add the axiom that the manifold should be Hausdorff ($T_2$) in each chart but in many fields, notably Lie groups, $T_2$ is enforced by other structure (the group laws) so this axiom is redundant here.
The easiest way to do this is to kit the manifold globally with a topology whose base is the open sets as reckonned by their images under the labellers, or, written backwards, the base for the topology is the collection of all preimages of sets open in $\mathbb{R}^N$ under the labellers.
Hopefully you can see that the notion of consistency as reckoned by overlapping charts, and thus the notion of the global manifold topology, is very much bound up in the physical concept of covariance and the Copernican notion that Nature's behaviour cannot depend on our mere description of it.
