There are some pretty straightforward examples here: Antisymmetric relation - Wikipedia, the free encyclopedia. If you don't understand them, tell us where you're having trouble, and we'll take it from there.
As far as the digraph of an antisymmetric relation is concerned, if a and b are two distinct points (nodes) on the digraph, and there's a directed link from a to b, then there can't be a link back from b to a. If there were a link back, then you would have a pair of elements a and b for which aRb and bRa without a being equal to b.
PS The answer to your second post: No.