"Waifu," which translates to "wife," is Anime Fanspeak for a fictional female character from an anime, manga, video game, etc. that somebody finds very attractive and sees as their ideal significant other.
Use of waifu as the Japanese transliteration of the English word "wife" dates back to at least the 1800s, but it began to be used more frequently starting in The '80s. It was adopted into English slang around the Turn of the Millennium. It's generally believed that this usage originates from Azumanga Daioh, where the teacher Mr. Kimura uses the phrase mai waifu—the Japanese approximation of the English "my wife"—to describe his own (real-life) spouse.
The term hasn't picked up the same connotation in Japanese—it only really exists as slang in the Western world—but the same concept of course exists in Japanese geekdom. Their equivalent is yome, or "bride."
"Husbando," i.e. "husband," is the Spear Counterpart for fictional male characters who cause this reaction in fans. This is entirely invented from how "waifu" is used in English slang; the proper Japanese transliteration of "husband" is hazubando.
For context, the actual Japanese word for wife is tsuma, and the actual Japanese word for husband is otto.
Compare and contrast with Perverse Sexual Lust, which is all about Real Life people being attracted to fictional characters, the latter of whom may be referred to as husbando or waifu. While they often overlap, "waifu" is etymologically unrelated to the English word "waif", typically used to refer to a thin person; it's not to be confused with tropes named after that term, like Glacier Waif and Waif-Fu (the latter is a pun on "waif" and "kung fu").