A person of any gender who has the potential to be attracted to both males and females. This does not mean that they are:
1) trying to attract attention
2) not-quite out of the closet as gay/lesbian
4) about to be unfaithful in any way.
A bisexual person has the potential to be attracted to either gender. This doesn't mean that they will be incomplete without regular sex with both genders; it's more of a potential energy thing rather than kinetic. It's also an emotional/romantic thing, not just about sex.
Although some members of the gay/lesbian community look down upon bisexuals, it's just as real and valid as any other sexual orientation.
Abigail had a crush on Naomi for a while, but then got together with Aaron, where she is perfectly happy and on her way to falling in love. She is bisexual.