~ the genealogist's labour of love is the anthropologist's headache
Conventionally, one's first cousin is one's parent's sibling's child. In predicate terms:
cousin-of ( ego ) = child-of ( sibling-of ( parent-of ( ego )))
Or, for the sake of convenience, psc. One's second cousin is one's parent's parent's sibling's child's child, or the grandchild of your grandparent's sibling (ppscc). One's third cousin would be pppsccc, ad infinitum. Cousins traced through your mother's line are called "matrilateral cousins;" those traced through your father's line are "patrilateral." As Webster 1913 indicates below, the word "cousin" originally etymologically referred specifically to one's matrilateral cousin.
While the basic number distance of cousinage – first cousin, second cousin, etc. – is based on generations through your ancestors (the p's), "removals" are counted on the basis of the cousin's ancestors (the c's). "Once removed" means that there's one generation's difference between yourself and the other, either forward or backward, except when that would be your aunt or uncle. For example, your first cousin once removed would be your first cousin's child, or pscc. It would not include your ps, who would be your aunt or uncle. Your second cousin once removed could be either your parent's cousin (ppsc), or your parent's cousin's grandchild (ppsccc). "Twice removed" means that there's two generation's difference, etc. Any "cousin" relationship in which the "p's" and "c's" aren't symmetrical is a "removal," based on the parental generation but "removed" by the difference of c generations from p generations.
Though I'm not entirely sure about this one, the obsolete term quater-cousin seems to have referred to anyone within four generations; this would include all your cousins out to your fourth cousins, plus your third cousins once removed, your second cousins once and twice removed, and your first cousin thrice, twice, and once removed. The term cater-cousin, I think, meant anyone even further related than that.
Euro-American culture is ambilineal — that is, we track descent through both parents. The vast majority of cultures (85%) are unilateral; that is to say, you are only actually descended from either your mother or your father (depending on the culture), but you're only related to the other. It's still a blood relation, but not "descent" per se.
Some of these cultures practice the customs of levirate, in which a widow marries her late husband's brother (à la Hamlet, minus the melodrama); or the sororate, in which a widower marries his late wife's sister. In these cultures, some of your "cousins" may be considered brothers and sisters. Furthermore, depending on the rules of descent and the customs controlling incest, you might be able to marry some of your first cousins and not others. For these reasons, most societies keep track of "cross cousins" and "parallel cousins."
A parallel cousin is the child of your parent's same-sex sibling. For example, your mother's sister's child is your matrilateral parallel cousin, and your father's brother's child is your patrilateral parallel cousin.
A cross cousin isn't just a cousin who's got a short temper; technically, a cross cousin is the child of your parent's other-sex sibling. So, your mother's brother's child is your matrilateral cross cousin, and your father's sister's child is your patrilateral cross cousin.
This isn't even to mention keeping track of step-cousins (cousins by marriage only) or half-cousins (cousins through an ancestor's half-sibling)! On top of all this, during the medieval era through to the early industrial era, the word "cousin" could just mean an old friend, or a friend of the family. As Webster 1913 indicates, it was also used as an additional title that could be conferred upon a peer, or else it was used to refer to an earl.
Though it has always been considered incestuous in the West to marry one's sibling, throughout most of our history (until the 20th Century) it was also considered incestuous to marry one's first cousin. Though it's still frowned upon, it's no longer illegal in most Western societies. In some cultures, the first cousin on one or the other side may be considered the partner of choice. Second cousins and further are considered by Western culture to be alright to marry.