The term remix does not, as you might surmise, mean that the engineer who mixed your song does it (mixes) again. Of course it could be anyone who mixes your song again, not just the person who originally did it, but - boy am I getting off the track here. The point is that the term remix has come to mean that some 3rd party has taken a popular song, typically and older one, and added new and different sounds to it, to create an alternate version of that song. In most cases the remixer only has the final version of the original song to work with. In other words, he/she does not have the original separate tracks - like the drum track, guitar track, bass track, vocal track etc. In fact, the remixer usually does not even have contact with the original artist, producer or publisher.
This, of course, is where copyright law comes into play. If you are going to get into remixing, you'd better make sure you either have a lawyer advising you, or you have a license from the original publisher or copyright owner. Lots of people hide behind the Fair Use Doctrine or claim parody protection. But that won't stop you getting sued if the copyright owner comes after you. So be very careful if you want to remix popular songs.
Anyway, assuming the copyright issues are covered, you can get some great ideas on how to do remixing form this article here: