The term is used as a very loose categorization, but is usually used to describe artists playing a style of metal which is considered either a unique approach to metal music or difficult to define as strictly metal or alternative. Faith No More is a good example of a band in which both criteria apply.
Heavy metal is an essential component of the music, but it was very different from the thrash underground of the 1980s. Initially alternative metal appealed mainly to alternative rock fans since virtually all 80s alt-metal bands had their roots in the American indie underground scene. Alt-metal bands commonly emerged from hardcore punk (Corrosion of Conformity), post-punk/gothic rock (Jane's Addiction), noise rock such as the "pigfuck" sound of Big Black and Sonic Youth (Helmet, White Zombie), grunge (The Melvins, Soundgarden), industrial music (Ministry, Nine Inch Nails), and other movements in the indie underground scene, although it was not uncommon for bands to incorporate a wide variety of influences (such as Soundgarden, who lists Bad Brains, Bauhaus, and the Butthole Surfers as major influences). These bands never formed a distinct movement or scene; rather they were bound by their incorporation of traditional metal influences and openness to experimenting with the form, usually by way of their eclectic influences and uncommon approaches. For example, Jane's Addiction utilized performance art and a bohemian aesthetic, Corrosion of Conformity, The Melvins and the now defunct grunge band Soundgarden had a fondness for subverting '70s metal, and Faith No More injected funk and rap music into their brand of alternative metal, while Primus incorperates funk, progressive rock, elements of thrash metal and punk rock, and an obscure Residents-esque touch in to their form of the genre.
The grunge movement of the early 1990s, which itself was a combination of 70's metal like Led Zeppelin and Black Sabbath and underground punk, helped increase the audience for such bands, and these artists were as comfortable playing to alternative rock fans on various Lollapalooza line-ups (itself founded by Jane's Addiction frontman Perry Farrell) as they were opening for metal bands like Metallica. With the changing of the musical landscape by the popular breakthrough of alternative rock, "alternative metal" became a new phrase used to describe bands in the early 1990s who managed to make relevant Nirvana era music that, as metal historian Ian Christe states, was "heavy without necessarily being metal". Newer bands emerged in this era with their distinctive takes on metal: White Zombie, Nine Inch Nails and Fear Factory started the industrial wave, combining techno-like beats and heavy guitars, Tool immersed itself in prog-rock influences, Rage Against the Machine was as informed by hip hop and post-punk agitprop such as Gang of Four as it was by metal, and Helmet molded a background in jazz and noise-rock/post-hardcore influences into a highly influential strand of intense rock music.
As the 90s progressed, alternative metal's sound became more standardized as newer bands drew inspiration for the same collective set of influences that included RATM, Korn, Nine Inch Nails, and Helmet. Helmet in particular, with its downtuned riffs and aggressive dissonance, created the sonic template for the nu metal movement. The chief distinctions between alternative metal and nu metal, aside from the generic sound, are the latter's tenuous (or even non-existant) connection to the underground rock scene and the DIY ethos that informed the musical approaches of past alternative metal bands, as well as the reluctancy of alternative metal bands to explicitly align themselves under the heavy metal banner.
Metalhead: They are alternative metal music. They are too wimpy to play real metal, so they play alternative rock pretending to be metal.