A female calf (and later, the resultant adult cow) that is sterile and shows masculine traits because it was the twin of a male calf.
"It is a fact known, and I believe almost universally understood, that when a cow brings forth two calves, and one of them a bull-calf the other to appearance a cow, that the cow-calf is unfit for propagation, but the bull-calf grows up into a very proper bull. Such a cow-calf is called in this country a free martin."
--John Hunter, 1786, Observations on Certain Parts of the Animal Oeconomy (sic)
In 95% of cases, cows pregnant with twins will join placentas1, causing blood to be shared between the two calves. In cases in which one calf is male and one female, the female will pick up male sexual hormones that disrupt healthy development of her genitals2. She will usually have regularly formed exterior genitals (although the clitoris may be enlarged), but she will likely have small and undifferentiated gonads, and a uterus and cervix that are only strands of connective tissue. In extreme case, the anus may also be fused closed, although this is rare. In many cases the underdevelopment is subtle enough that a blood test is necessary to identify the condition.3
Sometimes the male fetus will abort, leaving the female, still a freemartin, to be born alone, with no sign that she was once a twin. This is very rare; usually if one twin aborts, the other will too.
Freemartins may be slightly heavier and larger than other females born as twins, but not as large as single-born females. They also have higher levels of marbling in the meat, and the meat generally tends to be of higher quality. The development of teats seems to be unaffected. I have been able to find very little on the behavior of freemartins, except that Wikipedia notes that historically freemartins were used to detect when cows were coming into estrus; the freemartin will try to mount to the cow, at which point that cow would be removed to an isolated pen for breeding. All studies I can find deal with freemartins that have been injected with hormones in order to androgenize them, making the mounting behavior less interesting (in my opinion, anyway).
I have found references to freemartin sheep, goats, pigs, and, unsurprisingly, oxen and river buffalo. Freemartinism is much less frequent in sheep and pigs; unless otherwise specified, freemartin refers to a cow. I would assume that oxen and buffalo originally had twinning/freemartin rates similar to cattle, but in recent years the twinning rates in domestic cattle has been increasing. This seems to be largely due to a positive correlation between twinning and increased milk production. There are also some breeding programs that encourage twinning.
Freemartin comes from the words 'free' plus 'mart', a Scottish word meaning a beef animal fattened for slaughter (Webster's Third New International Dictionary). 'Freemartin', free-martin', and 'free martin' are all correct.
1. Fun techspeak: they form a chorioallantoic anastomosis, where the foetal portions of the placentas are joined.
2. It just so happens that the fusion of blood vessels generally happens after about a month of gestation, which is just before sexual differentiation commences. Male sexual differentiation starts a few days before that of the female. The female receives anti-Mullerian hormone from the male, which, as you might suspect, inhibits the growth of her Mullerian ducts. The exact day that the twins start sharing blood matters, as this is probably the primary determination as to how stunted the female's sexual organs will be. The external parts of her genitals are actually formed as part of the urinary tract, as opposed to the reproductive tract, which accounts for the (usually) minor differences in visible genitalia. As of yet we have no good explanation as to why the anti-Mullerian hormone does so much, but yet the steroid hormones of the male seem to have so little effect on the female.
3. Okay, obviously a blood test isn't going to tell you if her genitals are odd. When the placentas join, blood stem cells are shared between the twins, and therefor both of the twins have two genetically different bloods flowing in their veins (remember, these are fraternal twins, not identical). If a cow has male (XY) chromosomes in her white blood cells, she is a freemartin. This sharing of blood happens even in same sex twins, meaning that 95% of cattle twins are genetic chimeras. Cool, huh?