SIDs, or PSIDs, are data that have been extracted ("ripped") from C-64 programs. To play them, you need a SID-player. The most popular one is Michael Schwendt's Sidplay, which is available for a lot of different operating systems. If you want to listen to SIDs on a real C-64 (without having to cut off the header and writing an IRQ routine to play it) you can use Burglar's RealSIDPlay (download it from one of the usual FTP sites).
You may be wondering what the point of listening to C-64 music is. Well, you can listen to it just like any other music. SIDs have a very special sound, which might take some time getting used to, but if you get used to it you might start enjoying SIDs as much as I do. Listening to SIDs these days is also very economic and easy: the whole HVSC (see below) doesn't occupy more than 75 MB of disk space. Just download it and unzip it and you have a really great collection of SIDs, including the ones from all the old games and also tons of tunes from demos, music collections, intros and other programs. A SID doesn't need more disk space than a MIDI file, and it comes with all the sounds and a play routine built in. It also contains information about the composer, the title, copyright and some other stuff.
If you're really into sid music, you should take a look at the High Voltage Sid Collection. They've got more than 10000 songs in there. If you want to download them one by one, you can find all the sids from the High Voltage collection at some FTP sites, and you can find information about that too on the High Voltage home page.
You'll find a great search engine for the HVSC at Tomten's SID search.
You'll also need a program to play these songs for you. Get SidPlayer at The SIDplay Home page, so that you can listen to great music all the time... Oh, and don't forget to check out reSID there, it's a great improvement in sound quality! If you want the best sound quality, try RealSidPlay on a real C-64 (or in VICE, which uses reSID).
If you want to use playlists and stuff with Sidplay for Linux/Unix, take a look at sidtools, a small collection of utilities I've written. It lets you create a database that you can search for authors, titles, copyright information, load, init and play addresses, number of subtunes, default start tune and file size. The search engine produces play lists, that you can pipe directly to the list player or save as a file to play it later. You can also use directories as playlists, or play random tunes.
I thought I'd just tell you which my favourite sids are at the moment. This top 10 list is in no way absolute, and which ones are my favourites changes from day to day. These are really great tunes, anyway...
Most of these tunes are in the HVSC. The exceptions are Extasy, by DAF, which is used in Samar's demo Extasy, and Premature, by Glenn Rune Gallefoss, which is used in Domination issue 13. (Updated 20th December 1999.)
Okay, that wasn't easy... There are so many great tunes. Now flame me! :-)