Second cousin twice removed. Sounds mathematical, as though establishing a family relationship required the use of trigonometric algorithms. But like so many apparently daunting areas of expertise, it's really just a matter of learning the lingo. In the definitions below, "you" serve as the frame of reference, around which all the other relations revolve.
Uncle and Aunt
Your parents' brothers and sisters, and their spouses. You don't need any blood connection to be a full-fledged aunt or uncle. Your mother's sister's husband is not your uncle by marriage. He's your uncle, fair and square.
Niece and Nephew
Son and daughter of your siblings and siblings-in-law. As with uncle and aunt, they're not your nieces and nephews by marriage, they're your nieces and nephews, plain and simple, even if they're your spouse's siblings' kids and don't have a single drop of blood in common with you.
First Cousin, Full Cousin, Cousin-German
Children of your parents' brothers and sisters. You and your first cousins share one set of grandparents.
Double First Cousins
If a pair of brothers marries a pair of sisters, their kids are not only first cousins, they're double first cousins: They have both sets of grandparents in common.
You and the children of your parents' cousins are second cousins and share at least one great-grandparent. Your child and your cousin's child are second cousins.
You and the children of your parents' second cousins are third cousins and share at least one great-great-grandparent. And so on with the fourth, fifth, and sixth cousins.
First Cousin Once Removed
A relationship that is removed is one that exists in two different genealogical generations. Generation refers to the order of birth, a genealogical level. Your aunt and your mother may have been born 20 years apart, but they are still of the same generation. Your parent's first cousin is your first cousin once removed. The child of your first cousin is also your first cousin once removed: your grandparent is that child's great-grandparent. You can do the whole "removed" thing for every category of cousins—second cousin once removed, and so on. But by then you'll probably drive everyone completely crazy.
Grandaunt and Granduncle
We always called grandpa's sister "great-aunt." But the experts say that terminology is incorrect, and the terms above should be used instead to refer to the siblings of your grandparents. (Which doesn't mean we have any intention of using them.)
Great-Grandaunts and Great-Granduncles
Your great-grandparents' brothers and sisters.
Family by marriage: Your spouse's parents, spouses of your siblings, and spouses of your spouse's siblings. That is, your brother's wife is an in-law, but none of her siblings are. And your husband's sister's husband is your in-law, but none of his brothers are. And in-laws pretty much stop with your parents-in-law and your siblings-in-law. You are not in-laws with the parents of your sister-in-law's husband. And the two sets of parents of a couple are not in-laws to each other either; they are the competitive parents—an entirely different category.
Your husband or wife's blood relatives -- the in-laws that are biologically related to your spouse.
Birth Mother, Biological Mother, Natural Mother
Terms for the biological mother of a child who has (usually) been adopted by other parents.
Mother of a child who is not biologically her own.