You don't generally hear people asking what a penis is, but "what is a hymen?" isn't a ridiculous question. Men's nether regions are all right out there, but women's are all on the inside, all hidden and mysterious. You can't blame a guy for getting confused by all the parts and their names: vagina vs. vulva, clitoris vs. hymen, ovary vs. cervix. The hymen is a pretty basic part, though, and most guys who have gone through puberty have at least some idea of what a hymen is.
The hymen is a thin membrane that is at the entrance to the vagina. It usually doesn't completely cover the entrance, otherwise women wouldn't be able to menstruate. Occasionally, a girl has a condition called imperforate hymen, in which the membrane does stretch completely across the entrance. This is usually discovered when she has begun menstruating but not bleeding, and needs to be rectified quickly to prevent serious consequences. A doctor can generally make a small incision to tear the hymen enough to let blood out.
The purpose of the hymen isn't really clear. It is probably a leftover of fetal development of the vagina. It could be there to help keep dirt or other objects from entering an infant's or young girl's vagina. In fact, a torn hymen in a very young girl is sometimes a sign of sexual abuse.
Increasingly, some countries are giving "virginity tests" to women in an effort to "shame" them into refraining from sex before marriage. But these tests usually just consist of checking to see if the hymen is intact. The problem with this is that the hymen can tear easily. Activities such as horseback riding, cycling and strenuous sports can cause a girl's hymen to tear. So can using a tampon, masturbating or having a Pap smear. Some women are born without hymens. And on the other hand, sometimes women's hymens have such large passages through them or are so flexible, a woman who has had sex will appear to be a "virgin" using this standard.
When a woman with an intact hymen has sex for the first time, the general thought is that it is always painful and she bleeds. However, both, or either, or neither could occur. Hymens are as individual as the women who have them, and you can't make assumptions about their possible past activities based solely on a thin tissue in their bodies.