Mashups are most simply explained as a form of a remix.
Remixing is nothing new - talented DJs have been making dance versions of songs for decades. The basic idea is to make any song danceable, so there are club or dance versions of everything from Madonna, which you would expect, to "Stairway to Heaven," which you might not expect and which is a HORRIBLE idea.
I can't stress that enough. Please do not pursue finding that track, or worse yet, trying to create it on your own. Seriously.
Remixes and mashups are blurred when source material is augmented with new vocal tracks, beats and different songs. An early example of a remix that treaded heavily into mashup territory is DNA's placement of Suzanne Vega's a capella song "Tom's Diner" onto a beat from Soul II Soul, that became a huge international dance hit in 1990.
Done without Vega's knowledge or permission, it became such a massive underground hit that she embraced it. Her record company bought the song and released it. Eventually the song hit No. 2 on the UK charts.
Vega grew to love the remix, and subsequently oversaw the release of "Tom's Album" in which she included multiple remakes and remixes of the song from DNA and other producers. My favorite of that album is "Jeannie's Diner," by Mark Davis and Marilyn E. Whitelaw, which reworks the theme to "I Dream of Jeannie." I can't quite describe to you how awesome it is.
A weird side note to this whole "Tom's Diner" discussion is that the original a cappella track was used by MP3 developer Karlheinz Brandenburg to refine his audio compression technology and make sure his new format was staying true to the original music. This has earned Vega the title of "Mother of the MP3," bestowed upon her by nerds around the world. Please stand in awe of her, for she is glorious.
Andy Kline is the program director of KXLL/100.7 FM - Excellent Radio. He can be reached at firstname.lastname@example.org. To hear this track and a couple other "Tom's Diner" versions, tune into KXLL's "Remix at Six" Friday March 7, at 100.7.