A waifu is a female fictional character who has been loved by one person so much that that guy decides to marry her. That's right, he wants to marry a fictional character. This person must be a pathetic basement-dwelling loser or something, if he declares his love to something that doesn't exist in real life.
Most common targets to be waifu are cute girls from anime series (again because otakus). Often, though, waifus can also be Pokémon like Gardevoir or ponies like Twilight Sparkle (because some of them are furries too, and in the latter case bronies). Rarely you'll find someone who wants a live-action sitcom character to be his waifu, and probably for a good reason too.
There's a male equivalent to this: the husbando, often gushed over by girls and gay men.