Het is an acronym for Heterosexual and means any kind of fic that includes or evolves around a heterosexual relationship.
There is het in the fandom, but the lack of recurring female characters as well as the focus of the story on the relationship between the brothers (no matter if you interpret it as brotherly or incestuous) keeps this part of the fandom relatively small. Favorite pairings include canon pairings Sam/Jessica, Sam/Ruby, Sam/Sarah, Dean/Cassie, Dean/Lisa and of course, John/Mary, and fanon pairings such as Dean/Jess, John/Jo or Castiel/Anna. Crossovers with other fandoms mainly pair Dean with Faith or Buffy from Buffy the Vampire Slayer.
Another subset of het includes Winsister fic or AU stories involving one or more female Winchesters, often in place of either Sam or Dean. Sometimes these characters are female versions of Sam and/or Dean (Genderswap) while in other cases they are entirely original characters. Many, but not all, of these stories involving "het Wincest."
There is a strange phenomenon in which some fans strongly dislike het fic and will go so far as to say they are squicked by it - regardless of the type of sexual content (if any). The phrase "you must always warn for het" is often used sarcastically in relation to it, although it is used seriously by those fans keen to avoid any mention of heterosexuality.
The reasons behind this are diverse. Some fans simply do not wish to read about a character disrupting their Slash OTP. But there are many other elements at play ranging from a desire of heterosexual female readers to find escapism in Slash, or to not wish to read Het because of their own issues with sexuality or misogyny.
For more, see pages for specific pairings.