As perhaps the best example of a phenomenon that "looks" very different in high dimensions, I suggest knot theory. The first hint that there are surprises in store is the observation that knots in the usual sense (embeddings of S1S1 whose complement has non-trivial topology) do not exist in dimensions other than 3. However, there are analogous objects in higher dimensions: A kk sphere can be embedded in a k+2k+2 sphere to form a sort of "high dimensional knot". So the important thing is that the "knot" has codimension 2 in the ambient space. Try thinking about what these high dimensional knots look like :)