What is the fourth dimension of a Tesseract? Is the fourth dimension of the Tesseract time? Is that why it is represented as a moving 3D structure on Wikipedia? I am asking because I have trouble understanding what it is.
 A: It isn't time. The tesseract is an object in 4 (or higher) space dimensions. However, since we live in 3 dimensions, it is not possible for us to see a tesseract in all its glory. What we can see is the projection of a tesseract on a 3d plane. Now, depending on which plane is chosen for the projection, the tesseract looks different. 
The diagram on Wiki is trying to give you the best possible "view" of the tesseract by rotating the plane on which the projection is being done, thereby allowing you to see all possible projections of the tesseract. 
In fact, the text below the picture clearly states this 

A 3D projection of an 8-cell performing a simple rotation about a plane which bisects the figure from front-left to back-right and top to bottom

EDIT: Based on comments below, I'm adding the following notes - 
Firstly, a tesseract is a mathematical construct. It is a description of what a higher dimensional object would look like if such higher dimensions existed. Secondly, how do you conclude that you do not have such objects popping out of the 4th dimension, if it existed? You must remember that the world that you and I can see need not be the entire story. It may be true that higher dimensions exist but we can't access it. That is a totally logical possibility.
For instance, imagine an ant walking on a piece of paper. As far as the ant is concerned, it knows only of the 2 dimensions of the paper. Does that mean that the 3rd dimension that it cannot see doesn't exist? No! It simply means that the ant can only see and make sense of objects living on the paper or projections of three-dimensional objects on that paper. If the ant were smart enough, by studying the properties of the projected objects, it could infer the existence of the 3rd dimension without ever having seen it. The same logic applies to us. We will never be able to directly see the 4th dimension, but imprints of its existence might be present in the 3 dimensions that we can see. By properly understanding these, we can make progress. PS - String theorists believe that the world has 9 dimensions (+1 time dimension).
A: Simply, imagine some polystyrene balls (acting as points). Put one down. This is 0 dimensions. It cannot go anywhere. It is stationary. It has no properties. 
  Then, add $x$ balls next to it.It is now a line, or mathematically/physically, 1 dimension. You can travel forwards or backwards on it. 
In a perpendicular fashion, add more balls until it creates a square. This "pen" / "cage" is two dimensional. you can go left or right. Also, it carries the property of 1D ; letting you go backwards or forwards (That is why a square is called a '2D Shape').
IMAGINE a cube. It has many layers of squares on it. $x$ layers in fact. You are inside it. You can do anything you would normally do. Go up or down. But also, you are able to do anything a 2D person could do. So in 3D, you can go LEFT or RIGHT or FORWARDS or BACKWARDS. That is an example of YOU. 
 What about the fourth dimension? So, if you haven't noticed, the $d$ dimension equivalent of a cube would have $d$ edges coming off each of its vertices. e. g. A square has two edges coming off it at a vertice. Three for a cube. One for a line. And none for a point.

A: No, it's not time. Prahar is correct that it's basically the best way we can visualise it as 3D beings.
It's hard to describe in words, but if you're interested I would recommend reading Flatland: A Romance in Many Dimensions. It's a short story about a square trying to understand the 3D world, but it's easy to extrapolate the ideas to 3D and 4D (http://www.gutenberg.org/files/201/201-h/201-h.htm)
Using the analogy of 2D to 3D, imagine a being living on a piece of paper: Their world is 2D. How would this being see a sphere? They would see it as a series of 2D splices as it travels through the sheet of paper.
When we look at a Tesseract, we are basically doing the same thing: we can only see it as a series of 3D images changing with time, hence why it is animated.
