By "hypercube" most people mean "tesseract", although a tesseract is only one example of a hypercube (tesseract = 4 dimensional hypercube).
Let's look at it by analogy. A square is a 2-cube (2 dimensional hypercube). It has four sides consisting of straight lines. A cube is a 3-cube. It has six faces consisting of squares. A tesseract is a 4-cube which has eight "faces" consisting of 3-cubes.
Each corner (vertex) of a square has 2 lines. Each vertex of a 3-cube has 3 adjacent faces of the cube. Each vertex of a 4-cube has 4 adjacent cubes.
We can construct a 3-cube by folding a 2-dimensional grid of six squares into the 3rd dimension. Similarly, to construct a 4-cube we need to fold a collection of eight 3-cubes in the 4th dimension.
The problem with tesseracts is that we cannot visualise them in our 3D world. We can imagine various "shadows" though. If you shine a torch onto a 3-cube and look at its 2 dimensional shadow on a wall, the shape of the shadow might be a square, or some type of parallelogram. Similarly, if we could shine a "3D torch" onto a 4-cube and look at its "shadow" in our 3-space, we would see 2 joined and possibly distorted 3-cubes.
Each line in a square is at 90 degrees to the adjacent lines. Each square in a 3-cube is at 90 degrees to adjacent squares. Each 3-cube in a tesseract is at 90 degrees to adjacent 3-cubes. Remember that four 3-cubes meet at each vertex of the tesseract. All of them are mutually at right angles to each other. How is this possible? Answer: It is only possible in 4 dimensions (or more).
5-cubes, 6-cubes and even n-cubes are possible.