Parallels are variations of base cards that "run parallel" to the main set — same player, same design, but with a distinguishing physical feature that sets them apart. That feature could be a different border color, a unique texture, a special pattern, a different card material, an image variation, or a distinct printing technology.
Every product has its base set of commons, and then a smaller number of more limited parallels layered on top. The rarer the parallel, the more valuable it tends to be. Chasing parallels — especially color-coded or numbered ones — is one of the most addictive parts of the hobby. Once you start chasing the rainbow, there's no turning back.