The term hypercube is normally used to refer to a 4-cube, or four-dimensional n-cube. It's also often called a tesseract. Sometimes, more in line with mathematical naming conventions, it is used to signify a generalised n-cube, not necessarily of 4 dimensions.

The smaller (lower dimension) n-cubes are a point (0-cube), a line segment (1-cube), a square (2-cube), and the infamous "cube" itself (or 3-cube, in this terminology). All of these fit in 3 or less dimensions, so they are easy to visualise In fact, you can easily build or draw them all... But you'll never actually build or experience a hypercube. And I rather doubt you can trick your brain into "seeing" a hypercube in the exact same sense you can see (or imagine seeing) a cube.

Now, from a formal point of view, the definition(s) under n-cube are pretty much all you need to know. But it's always better to be able to "visualise" a mathematical concept. Here are a few ways to get started (we'll do just the 4-dimensional cube):

• Pretend you are a two-dimensional being (a la flatland) and a 3d cube passes through your world. What do you see? A succession of "slices" of the cube: just squares, if it goes through face-on, or a gradually shifting, expanding and collapsing polygon, if you're seeing consecutive cross sections at some tilt. Now remember you live in three dimensions, and imagine a hypercube floating through our space. You see a shifting 3d cross-section of the "larger" cube...
• Well, OK, but that's not a very exact way to imagine the hypercube. Another way to do it is to examine the progression through lower dimensions. A 1-cube (segment) is obtained from two 0-cubes (points) by placing one point at a distance of (say) 1 from the other, and connecting them with an edge (unsurprisingly!). A 2-cube (square) is obtained from two 1-cubes (of length 1) by placing them parallel at a distance of 1 from each other in a direction perpendicular to the 1-spaces (lines) which contain them, and connecting matching vertices with edges. Now, to get a 3-cube from 2 2-cubes you do the same: place them parallel at a distance of 1 from each other in a direction perpendicular to the 2-spaces (planes) which contain them and connecting matching vertices with edges. To get a 4-cube you have to do the same, with two parallel cubes, separated along some fourth axis, and with all pairs of matching vertices connects. Just believe for a moment that such a direction exists.
• Topologically, you get an exact (but somewhat confusing) description of a hypercube, by drawing a small (3d) cube inside a large cube (oriented the same way), and connecting matching vertices with edges. For the moments, just think of a wireframe model in three dimensions (with 12 edges on the outer cube, 12 on the inner, and 8 diagonal ones linking corners). Note that the space between the two cubes is divided into 6 regions by the trapezoids formed between wires. Now, if you pretend the vertices, edges, and faces (any quadrilaterals formed by the edges) of the model are the vertices, edges, and faces of your hypercube, and the 7 spaces enclosed by faces plus the space outside the outer cube are the 3-dimensional faces of the hypercube, then all the adjacencies and similar properties turn out just right! Many web pages and screensavers display rotating wireframes of this sort (with a "fourth dimension" rotation as well). They're pretty, but probably won't help you much. Just search for hypercube -- and maybe Java -- on Google.
• If you cut along some of the edges of a cube's surface (one made of paper, say), you get a surface you can unfold and lay flat. Depending on exactly how you do it, you will get something like four squares (former faces) in a row, with another 2 squares adjacent to the row, one above it, and one below. (If you've never done this, or at least folded up such a plan into a cube, try doing this in your head, or for real.) Note that if you know which edge attaches to which other edge, you can imagine exactly what life looks like to a flatlander who lives on the cube's surface; simply put, if you go "off the edge", you "reappear" at the corresponding edge. Now, what happens if you go through a similar process with a hypercube, cutting along two dimensional faces (squares), until you get something you can "lay flat" in three dimensions...? Some mental contortions might convince you that what you get is -- similarly enough -- four cubes in a row, with another 4 cubes stuck on, one on each side of the row. With similar "glueing instructions", you've got a perfect description of the 3d surface of a hypercube.

For the record: a (four dimensional) hypercube has (24) 16 vertices, at each of which 4 edges meet (one along each axis). That makes 32 edges (16*4, but that way you count each edge twice -- once for each edge). It has (2*4) 8 three dimensional faces, and therefore also 24 two dimensional (square) faces (8*6=48, as each cube must have 6 faces, but that's counting every square twice -- once for the cube on each side).