Steps

1. Select the right track. This is very important, since this is a derivative art form (you can't remix a nonexistent piece of music). You'll need, at the very least, a complete mixdown of the original track (taken directly from the CD). If you can get separate tracks (especially for vocals) directly from the recording artist, it will make your remix better, and your work easier.
2. Try to identify what you will contribute to the track. This can range from changing the feel by adding new rhythm tracks, to total destruction. Think about what sections you like best; what will you keep intact, and what will you change? This is an important stage, because it will shape your workflow throughout the project.
3. Dissect the track. Take the materials you have to work with, and do most of the slicing and trimming up front. You can do this in an audio editing suite, especially when it comes to cutting loops (see below for tips and links).
4. Experiment! Try all the available effects in your DAW/audio editing software to see how they will sound on each part! There are plenty of things to choose from ... delay, phaser, chorus, flanger, filters and other eq, reverb, amplitude modulation, ring modulation, frequency modulation, timestretching, pitch shifting or correction, vocoding and more.
5. Reconstruct (remix). First, set the BPM (tempo - beats per minute) and time signature (usually 4/4 in popular music) in your looping software. Next, import your loops. Once they are imported and time corrected, you should be able to choose any tempo you like, with very little loss of quality. (Note: If you are using Ableton Live, be sure to select a time correction method that jives with your sample type. Beat mode is fine for drums, but may not be great for vocals. Texture mode is fine for many samples, but will often affect the pitch of the sample slightly. Tone is usually good all around.) Now you can start to reconstruct the track. A safe and easy way would be to follow the form of the original (intro, verse, chorus, verse, bridge, chorus) but you can also completely change it and make it your own. You can layer the vocals from the verse over a portion of the chorus. You can take a verse as-is, cut individual measure of vocals, and superimpose them reversed. You can reharmonize the vocal or lead lines by introducing completely different elements. Have fun, and experiment!
6. Export your creation (mastering). When your remix now has a start and finish, and you are satisfied with it, you should export. Save all or export to a WAV or AIFF file (don't encode an MP3 just yet). Load this into your audio editing software, and normalize it to 99%. This will ensure that your levels at their highest point reach almost the maximum volume.
7. Distribute your remix. Convert this file to MP3, using Lame, or your favorite MP3 converter, and send it all around!

Tips

• Watch your quality settings when you convert. 128 is almost the standard, but it is noticeable. At least 160 or 192 is recommended. But remember the higher the bit rate, the larger the file.
• Remixes show up in nearly all styles. In the pop world, it is usually a functional thing rather than an expressive one - converting pop or rock tunes to be club-ready. The important thing, whether in dub reggae, hiphop remixes, house remixes of pop tunes, or whatever, is that the remixer adds their own personal touch to the track - bringing some important elements of the original, while adding their own recognizable style.
• Cutting loops is a fairly simple process. First, listen to your file, and identify the sections you'd like to cut. Then, select the phrase you'd like in your audio editing software, making sure to grab complete measures. A way to test your cut is to loop playback on the selection. If it sounds jumpy at the loop point, you may be selecting too much, or too little. Watch out for loops that include reverb tails or cymbal crashes, since these will often extend past the end of a phrase. Making sure your loops are cut precisely will make tempo correction within your looping software more accurate. In programs like Sonar and Acid, which use nearly the same correction methods, this is imperative. If you are using Ableton Live, you can work very easily with completely raw samples, and eliminate some of this. Ableton is easily the most flexible looping software on the market. It allows for many different types of granular-based pitch and time correction, variable start and loop points, and an easy graphic interface for time correction. Time correction is achieved by either specifying the BPM of the loop (often detected automatically), or by inserting markers in the loop inspection window, to indicate where each beat falls. All of this will accomplish the same result as cutting and looping, while still preserving the original file. You can also take this time to do some processing on your loops. If you only have the full mixdown, you can bring out vocals or individual instruments somewhat with EQ. Keep in mind that there is no way to completely isolate a single instrument or voice after a mixdown. For example, you can lighten up the lower register (kick, toms) and basslines by rolling off the low end. This will keep things from getting muddy, if you use the vocals from that loop over a new bassline, or new drums.