Most music mp3 downloads are illegal because of the copyright on the music as well as the licensing. Most popular songs are not licensed to be freely distributed - without paying for it. This is piracy. There are several services which allow you to download single songs rather than full albums - one of which is iTunes, and will generally cost you around .99 per song.
There are some free services that allow you to stream music, such as Pandora, Spotify, and iTunes Radio. Also, some more reputable music stores such as Amazon and iTunes also run specials where some songs are free if you download them during a specified time.
There are some music files that are released for free downloading on the Internet. These are generally songs that are not popular or mainstream. Check with the licensing and copyrights of the song before downloading to be sure.
As a final note: Try looking on the artist's official website or their record label's official site for any sneak peeks or any free downloads for their music. If the song is not available there, most likely it is not free to download anywhere else.