From Word Net Search: Boob

Noun S: (n) dumbbell, dummy, dope, boob, booby, pinhead (an ignorant or foolish person) S: (n) breast, bosom, knocker, boob, tit, titty (either of two soft fleshy milk-secreting glandular organs on the chest of a woman) Verb S: (v) drop the ball, sin, blunder, boob, goof (commit a faux pas or a fault or make a serious mistake) "I blundered during the job interview"

(I was not aware of the verb form, but it follows from the first noun form)

Boob could also be used to refer to a person who is acting like a clown, or it could be used to refer to something that induces cluelessness. We have a nickname for television in the US (although it's going out of fashion) called the boob-tube. I assure you it's not because it shows porn.

Noun S: (n) breast, bosom, knocker, boob, tit, titty (either of two soft fleshy milk-secreting glandular organs on the chest of a woman) S: (n) nipple, mammilla, mamilla, pap, teat, tit (the small projection of a mammary gland) S: (n) titmouse, tit (small insectivorous birds)

(I was aware of titmouse but not a small bird called tit)

Tits is generally the word for mammary glands, being a corruption from teat. That's what @ghoppe was onto with his, where you emphasize the pointy bit of the end of the mammary gland. It's more evident on a dog or pig, where there are a line of teats (or tits). The fact that this usage has spread to also refer to women is likely from the fact that country/urban life has comingled more of late than of years past.

The usage to associate tits with boobs however, is purely sexist. In polite society you should mention neither, referring instead to the bosom that @Robusto mentions, or breast, but only when you must. Bosom indicates "that feeling of your grandmother wrapping her arms around you when you're scared" and breasts is generally seen as more clinical in nature.

Generally in polite society you would just avoid references to a woman's breasts, and let it go.

When breasts are being discussed (such as in the context of bra fitting), women will often use the word "boobs" amongst themselves.

With men and men alone, tits is fine.

NOTE: I edited in response to a comment and decided I would give a little more attribution.