A remix involves taking elements of an existing song, like the hook, bass line, vocals or harmonies, and changing them or adding new ones. If you hear a tune and think that it would sound better with a different beat or edit, this is a good opportunity to try creating a remix. It is not uncommon for remixes of songs to become even more popular than the original tracks. Many DJs became famous because they remixed a popular track and capitalized on that song's success with their own unique style.
A mash-up is simply taking two or more songs and putting them together to create a new song. To make a mash-up, you might try taking the bass line of one song, the vocals of another, the hook of another, and the harmonies of yet another song and mash them all together into one new song. Mash-ups are becoming more and more popular among club DJs, and producers are even marketing albums that contain nothing but mash-ups.
Most DJs produce compilations, also called mixes or sets, of their long mixes, where songs overlap and blend into each other. In a megamix, the mixing happens quickly, and each song is played for only a short amount of time. For a set or full-length mix, each song plays almost in its entirety, and the DJ transitions smoothly from one song to the next.