"The number associated with your cousin has to do with how many generations away your common ancestor is. For example:
First cousins share a grandparent (2 generations)
Second cousins share a great-grandparent (3 generations)
Third cousins share a great-great-grandparent(4 generations)
Fourth cousins share a 3rd-great grandparent (5 generations)"
The link also explains the once removed thing. Which is basically the difference if the generations back to the common ancestor are different. Bob's relationship to a child of Alice would be 2nd cousins once removed since Alice's child is one more generation away (3 hops) from the common ancestor than Bob (only 2 hops).