How to understand Extra Dimensions? I don't understand how there can be extra dimensions. I've heard it explained to imagine there's a tiny door that we can't perceive, and when you do discover it and walk in, you have discovered a new dimension. But this doesn't make sense to me.
As an example, say I'm standing on the ground at (0,0,0). It looks like I'm on solid ground. Well, there's actually a tiny trapdoor below me that a dust mite is crawling through, and getting directly below me. So now it is at (0,-1, 0). It's not like I've added another dimension to the world, it's essentially just my eyes aren't good enough and I'm not small enough. How can you ever have more than the basic movements of front/back, left/right, and up/down?
 A: Imagine a particle constrained to move along the surface of an infinitely long cylinder of radius $R$.  When we zoom in close to the cylinder, this motion is clearly 2-dimensional, since the particle can move both along the axis of the cylinder and around the cylinder's circumference.

However, if we zoom out to distances far, far larger than $R$, then the motion starts to look 1-dimensional. When the circumference is extremely small, then motion around the circumference is very difficult to resolve - so for example, a particle which doesn't move along the axis but simply winds around the cirumference in circular loops will look as though it's basically sitting still.

A point on the surface of the cylinder can be labeled by two numbers - a real number $z$ which specifies a point along the axis of the cylinder, and an angle $\theta$ which specifies position around the circumference.  Near the cylinder, the motion is clearly 2-dimensional, but at large distances, motion in the $\theta$ direction is impossible to see, and so for such distant observers, the only dynamical degree of freedom for the particle would be $z$ and the motion would be effectively 1-dimensional.
Our limited primate brains are not capable of directly visualizing it, but from a mathematical perspective it is trivially easy to extend this idea to higher dimensions.  We might imagine, for example, that the position of a particle is labeled by four numbers - $x,y,z$, and $\theta$.  The first three coordinates can take any real number as a value, but $\theta$ is once again restrict to an angle between $0$ and $360^\circ$.  Loosely, you can imagine a tiny loop of radius $R$ at each point $(x,y,z)$. Just as before, at distances vastly larger than $R$, motion along the loop is impossible to resolve, and for distant observers it looks like the position of a particle is specified completely by $x,y,$ and $z$ alone.  The motion along the loop only becomes apparent when we use probes which are sensitive to distances on the order of $R$.

Through such compact extra dimensions, it is possible to imagine a universe with (possibly many) more than the three "large" dimensions that we are intuitively familiar with, such that the compact extra dimensions are simply unobservable on the length scales we've been able to experimentally probe so far. If we postulate that ordinary matter simply cannot propagate in these additional dimensions, then it's possible that they aren't even particularly small.  However, because gravity is intimately related with the structure of space and spacetime, it's possible that these otherwise-unobservable extra dimensions could have an important impact on gravitational interactions. In particular, they potentially go a long way toward explaining why gravity is (apparently) so phenomenally weak compared to the other elementary forces.
A: Let's begin by saying that human intuition stops at three dimensions. We have intuitive understanding of 1, 2 and 3 dimensions, but anything higher than that must be understood by analogy. It turns out that the human brain is very good with analogies and can make up for the lack of physical intuition.
Let's start with 0 dimensional space: a point. There is nowhere to go, we are stuck in place. "Size" in zero dimensions is zero, the size of a point. Add one dimension and we give the point the freedom to move forward or backward. Still no up/down or left/right. "Size" in 1d is the length of a segment, mathematically,
$$L = \int dx$$
2-dimensional space adds left/right, but still no up/down. Here "size" is area:
$$A = \int\int dx\, dy$$
In 3-dimensional space we add up/down. Here "size" is volume:
$$V = \int\int\int dx\, dy\, dz$$
Now you get the idea: even through we cannot experience four dimensions, it's only logical to say that "size" is the "four-dimensional volume"
$$\mathcal V = \int\int\int\int dx\,dy\,dz\, dw$$
And so on. The "door" you've heard about is meant to convey that we gain a new degree of freedom as in a new independent variable, from $x$ to $(x,y)$ to $(x,y,z)$, to $(x,y,z\cdots)$
There are more analogies: a line is to a point what area is to a line; volume is to area what area is to line; $N$-dimensional size is to $N-1$ dimensional size what volume is to area, or area is to line, or line is to point. And so on.
We cannot visualize  these higher dimensions but they exist in our head as mental constructs and we can manipulate them mathematically. These manipulations will feel more natural if we continue to run these analogies in our head.
A: Loosely, the dimension of a space is the number of coordinates required to label all the points within it. You can label spatial points in your room with 3 numbers (e.g. x, y, z), the points on the surface of a globe with 2 numbers (longitude and latitude), and the points on a circle with 1 number. Generally mathematical spaces may be contrived with any arbitrary number of dimensions that are incredibly difficult (if not impossible) to visualize.
According to general relativity, we live in a 4-dimensional universe of events: 3-spatial dimensions and 1 time dimension. The 3 spatial coordinates label where the event happened, and the 1 time coordinate labels when that event happened. For example, the candle on your nightstand going out was an event at (x,y,z,t) = (2ft, 3ft, 1ft, 11:59pm).
Some physical theories, such as string theory, use more than 4 dimensions, but may be dimensionally reduced through compactification. Compactification can also reduce physical problems to fewer than 4 dimensions.
