Family trees are complicated enough, but trying to put a family tree into words can be even more difficult. Do you know your second cousin from your first cousin, once removed? Do you know what to call your grandparents' niece? And why can you only remove cousins and never that little brat of a brother?
Ordinal notations like first, second, and third cousins simply denote the number of generations you have to go back before two people of the same generation find a pair of common ancestors. First cousins (or just cousins) share the same grandparents. Second cousins share the same great-grandparents. And so on.
If you have only one common ancestor, instead of a pair, you normally just add "half" to the description. For example, if your grandmother had sons by two different men, they would be half-brothers. Your father's half-brother's children would be your half first cousins.
"Removing" cousins from your family tree doesn't involve lopping off a branch. The word removed simply denotes a generational difference. Your first cousin, once removed, is your parent's first-cousin.
Putting It All Together
Here's how you can figure out how two people, let's called them Jerry and Gina, are related. First, find the ancestor that they have in common (whom we'll call Edna). Then, count the number of ancestors between Jerry and Edna and between Gina and Edna, not including Jerry, Gina, or Edna in the count. For example, if Edna is Jerry's great-grandmother, the number of ancestor between them would be 2: a parent and a grandparent.
If the two numbers you come up with are the same, then Jerry and Gina are in the same generation, so that number represents which "number" of cousin they are. In the previous example, they would be second cousins.
If the two numbers are different, the lower number represents the "number" of cousin, and the difference between the two numbers is how "removed" those cousins are. For instance, if Jerry has two ancestors between him and Edna, and Gina has five ancestors between her and Edna, Jerry and Gina would be second cousins, thrice (three times) removed.
This assumes that the children of Edna that Jerry and Gina descend from were both fathered by the same man. If that isn't the case, they are half second cousins, thrice removed.
With a little practice, naming your relationship to people in your family tree can become easy . . . until you start trying to figure in marriages between people who are already blood related. For example, if two men marry two sisters, their children will be double first cousins because they will share both sets of grandparents.
If you want a real challenge, take a look at the family trees of European royals from 200 and 300 years ago. Inbreeding among royalty was not uncommon at that time, and it led to both physical and mental derangements. For example, the mother of Spain's Charles II was also his first cousin, his paternal great-grandmother was also his maternal great-great-grandmother, and his maternal great-grandmother was also his first cousin, three and four times removed! This knotting of his family tree caused him to have both physical and mental deformities and rendered him completely sterile, which is how he became the last of the Spanish Hapsburgs.