When is it suitable to use background music on a web site? Do users like this?
In general I dislike background music on a website. It slows down load time, it surprises users who have their speakers turned on, and its annoying. It is similar to flashing text on a web page.
The only time it makes sense to me is for a site about music or for a band. Then it can be expected.
The better solution is to have a player on you web page like @aslum suggested that way people can choose to play your music or not.
Never. Well maybe if it's a myspace page. Though in that case it's more expected then suitable. Or if it's a website for a band, you could have music that the user could click to play, but it shouldn't automatically start, and there should be a way to stop it as well. Or if it's a game, you can have music, but again there should be a way to stop the music.
Almost no one likes having their speakers hijacked, especially if they are already listening to music (pretty common) having new music play over top is just plain annoying.
On sites that the user visits expecting music, either as primary (websites of bands, labels, individual music-related products, such as Original Soundtracks etc) or as secondary yet required content (sites promoting movies, computer games, musical instruments and equipment etc.) Even in these cases autoplaying music and/or video is discouraged. You may want to play a relatively non-intrusive (white-noise like), enticing soundscape (rain, ominous ambience etc) instead - but leave the decision of actually starting music/video to your users.
Otherwise the best answer seems to be "never"... at least never with autoplay.
Beyond the exception of band/musician websites, there are interactive stories or other interactive hyperlinked art websites where background music makes sense as part of the experience. However, as a general guideline, it's best to present a landing page describing what the user is about to embark upon WITHOUT music before exploring deeper into the site. Additionally, one should provide some easy mechanism to turn off or change the volume of the music on a given page. My experience is that when music plays without some sort of 'opt-in', it's generally annoying to the majority of users (as the responses here indicate).
If there is an exception to the rule it would be websites that music is expected such as for a band/musician. Ideally in these cases though you will put a very obvious media player on the page and give the user an opportunity to turn the sound off (and possibly even opt of the sound for future visits).
I have a music review web site, so when I'm online I'm always listening to some album, and have to visit band pages. Having the web site automatically play music is one of the things that irritate me the most (one hell of a mess when you're already listening to something else.) The background music should be an "OPT-IN" (not opt-out, not good enough) feature on web sites (I've seen a couple of those.) If I want to hear a band's music, I expect them to have a "samples" section where I can go voluntarily.
The same applies to those company web sites that have some spokesperson start chatting as soon as we land on their page. That's extremely irritating. Web sites should be silent unless I authorize them to talk/play music to me, period.