The third dimension is how we see and experience the world around us, where otherwise two-dimensional objects have depth.
So I can't really fully encapsulate what the third dimension looks like on this website because it's limited to displaying 2D, but look at the diagram and use those imagination skills to pretend it's 3D, and then drag me around in that 3D space. The coordinates to describe my location now would be the x coordinate, y coordinate and z coordinate. I can go forward, backwards, up, down, left, and right in this space, just like you!!
In terms of a four-dimensional black hole, the third dimension is the brane we humans are confined to, which is why we do not experience time dilation for ourselves.