In modern usage, a cousin shares a grandparent with another person, AKA a first cousin. The term is also used for a person who shares any common ancestor though it would not usually be used in place of aunt, uncle, nephew, niece and other terms for close familial relationships.
In colonial America, a cousin could be a nephew, a niece, the spouse of a nephew or niece, or any person who shares a common ancestor.
Second cousins are related through common great-grandparents. Third cousins are related through common great-great-grandparents, etc.
Once removed, twice removed, etc., are used to indicate that the cousins are not the same number of generations from the common ancestor. If the common ancestor is a grandparent of one cousin, but the great-grandparent of the other, then there is a one-generation difference and the two people are 1st cousins once removed.