A good first stop to hear a specific song or artist is YouTube, as there are a surprising number and range of artists depicted there. There are a good deal of label-driven videos, as well as independent homages set to music. Whether you are looking for a long ago popular band or something brand new, its likely that you will find some of what you are looking for here and will be able to get a feel for the artist's music and style.
Pandora Internet radio, AOL radio, Imeem and Yahoo! music all provide new and back catalog music for full length listening or in clips at no cost. Most have sophisticated search functions that can predict related or other bands you might enjoy and suggests them. Many of these host video music and interview content as well that gives you a better window on a given band.
Many online retail outlets that sell music have a music preview function to help you decide whether or not to make a purchase. These come in really handy when you know the band, but don't know the name of the song and can fairly quickly figure it out by the process of elimination. Some require you to sign up for an account with an email address or to have a certain music player software installed, but they are free of charge and can be a handy preview and listening tool. Amazon.com, iTunes and Borders Books and Music all use this feature.
Increasingly, major record labels, and even independent artists are building sophisticated web fronts to preview, distribute and sell their music. In some cases, these sites are used to market entire new albums, greatest hits or entire song catalogs. Songs and videos are often available for listening and viewing at their full length free of charge without the clutter and interference of ads or a hard sell atmosphere.