Remixes are today such an essential part of promoting a single that you can tell if something is a hit just by looking at the names in the brackets like Morales, Basoski, Calderone, Negro, Tiesto, Maas, Hybrid.... And there is nothing more artistically challenging than using something that already is a hit and turning it into something even better. There are, of course, issues with bootleg remixes that are actually dubs of the original version, but I will get to that a bit later.
The point of it all is that both the author and the remixer benefit from the remix: the author still has the copyright and all that goes with it and gets more airplay (even if the original is long forgotten, or did not do well upon its first release), and the remixer gets to reap some of the benefits of having a hit without the hassle of doing everything from scratch. This is important because many aspiring artists do not bother to ask for permission to do a remix, but simply cut up the track, do a new layer of drums or a new bass line over the original and slap on the words "so-and-so’s remix", thus negating the fore mentioned concept of a remix being a mutual thing. Moreover, this bootlegged version of the song can only be called a dub (if anything), but certainly not a remix. There is some dispute over whether bootlegged "remixes" are a free way of promoting your published song, with the idea being that the people who, for instance, download the bootleg "remix", will be so kind as to buy the original, but that seems as utopian as thinking the "remixer" will not sample your original and publish it as his or her own work.
Doing a good remix basically comes down to getting the samples from the author (the legitimate way), keeping the focal points of the original and adding your own flavour to it. Everything else is up to the remixer and his or her artistic and technical ability to produce a good track. Here are some details on what to do and how to do it:
Step 1: getting the samples of the original from the author
Most producers today keep their original samples. You will need to contact the author of the song you want to remix and ask for the samples along with a written permission to do the remix. This is the most boring and stressful part of the process but the legally most important. A label that holds the copyright or a person holding the copyright will then choose whether to give you the samples or not. The problem is that most people want to go for the big ones, but the big ones want the bigger ones to do the remix, and there we go back to mutual interest.
Luckily, today many aspiring labels need remixes of their artists' work for promotion (remember the brackets I mentioned before) so they offer downloadable samplepacks for people to remix. There is a disclaimer on the page (traditionally long and boring as legal disclaimers tend to be) about them holding the copyright and you getting the promotion you want only through them (which is fine, since otherwise, they would be contacting you as the bigger one). After downloading the samplepack (generally in .zip or .mp3 format), you will need to convert the file in order to be able to use it (most commonly into a .wav format). If the samplepack is already in the desired format you can go right on to Step 2.
Now that the boring legal part is over and you have the samples, you can get down to business.
Step 2: keeping the focal points of the original
A good remix has all the good things the original does, and more. But it is as important to know what the good features of the original are, as it is to put your own samples in. This means that you will need to listen to the original at least a dozen times in order to hear (or feel) where the focal points are: it could be the vocal highpoints, the bass line, the melody, the instruments used, the chords, the filtering...Just be careful not to keep all of those things, because you will be left with an overproduction of the original.
One thing that helps with this step is the samplepack. Most of the samples will either sound totally right or totally wrong for the remix, but remember that first impressions aren't something to go by. This means you need to get to know the samples by listening to them, again, at least a dozen times.
By now you have either lost all will to do a remix, or you are ready for the great part.
Step 3: adding your own flavour
You can tell a good remixer by his ability to use the focal points of the original, and yet totally change everything else to suit his own memorable style. This is where your creativity comes in. Experiment with the focal points, rearrange them, filter them, cut them up, do anything and everything to them and you will be inspired (hopefully) to add something you think would go well with them.
How you produce your own samples is totally up to you, but the important thing is that you do produce them. I suggest using software synthesizers because of their beatmaching abilities and software samplers because of their precision and ease of use, but if you are more used to producing on hardware equivalents, go right on and do it that way. You will also need an editor to mix down all the elements. I suggest Sound Forge, Cool Edit or similar sound editing software, but whatever you are used to is, of course, also OK.
After you have produced your remix, you will want the people to hear it, but that is a topic for a whole new feature. To sum things up, there are a lot of labels and artists out there that need remixes and a lot of artists who want to do remixes for them, and if it is in both their interest, a remix is bound to be done. Samples of the original are necessary for the remixer to be able to use the focal points of the original as well as a means of inspiration for producing his or her own samples that will add the specific flavour, making the newly produced combination of the two a true remix – something the listeners will know after only the first couple of beats.