What are the differences between first, second and third cousins, and once, twice, etc. removed cousins?

First, second and third without any removal means that you are the same generation as them, and the degree refers to who was your common ancestor.
- Your first cousins share one set of grandparents as yourself (ie. your father and your cousin's father are brothers).
- Your second cousins share one set of great-grandparents as yourself (ie. your grandfather and your second cousin's grandfather were brothers).
- Your third cousins share one set of great-great-grandparents as yourself (ie. your great-grandfather and your third cousin's great-grandfather were brothers)

Removal indicates how many generations higher or lower you are relative to your cousin. For example, first cousin once removed refers to both your parent's first cousin, and your first cousin's child. In this way, that child will also consider you first cousin once removed, as you are his parent's first cousin.

The way I track my relatives are as follows (I'll follow relatives according to male relatives to keep it simple):

Father's first cousin = first cousin, once removed
Grandfather's first cousin = first cousin, twice removed
Great-grandfather's first cousin = first cousin, thrice removed

First cousin's son = first cousin, once removed (your child and your cousin's child are second cousins)
First cousin's grandson = first cousin, twice removed (your grandchild and your cousin's grandchild are third cousins)
First cousin's great-grandson = first cousin, thrice removed (your great-grandchild and your cousin's great-grandchild are fourth cousins)

Father's second cousin = second cousin, once removed
Grandfather's second cousin = second cousin, twice removed
Great-grandfather's second cousin = second cousin, thrice removed

Second cousin's son = second cousin, once removed
Second cousin's grandson = second cousin, twice removed
Second cousin's great-grandson = second cousin, thrice temoved

A seventh cousin four times removed is the seventh cousin of your great-great-grandfather (who is four generations higher than yourself) or the great-great-grandson of your seventh cousin. Basically, to think of it logically, Xth cousin Zth removed means the Xth cousin of your direct ancestor Z generations upwards; or your Xth cousin's descendent Z generations downwards.

For example, when someone says George Washington is his fifth cousin, eight times removed, that means that one of his ancestors eight generations ago (eg. his g-g-g-g-g-g-grandfather) was a fifth cousin of George Washington! Not very exciting when you know how distant that is.

The first number refers to how far back you must go to a common ancestor from the elder of the two cousins. For example, 1st cousins must jump one generation to get to a common ancestor (i.e. the grandparents):

Grandmother
| |
Mom Aunt
| |
Me 1st cousin


While 2nd cousins must jump two generations to get to the common ancestor:

Great Grandmother
| |
Grandmother Great Aunt
| |
Mom 1st cousin, once removed
| |
Me 2nd cousin


Note that the "1st cousin, once removed" only has one jump to the common ancestor (hence the 1st), but is one generation removed from me (hence the once-removed).