It's a word put together from Chillin' and Relaxing. It actually means the same. If somebody should cool down you could say he or she has to relax or has to chill out. If you decide not to choose between these to, or you haven't made a decision while you already have to say the word, you say 'Chillax'.
Term coined by Carles of Hipster Runoff referring to a style of music that emerged in Summer 2K9 primarily from the Brooklyn music scene.
Chillwave generally features a fuzzy sound with a slow but steady pulse-beat that hearkens back to 1980's pop culture; it may sound like the radio being played in the car during an episode of Miami Vice recorded on a crappy VHS tape and played back after said tape's been melting in the attic for a few years.
Although it can be either electronic or markedly lo-fi, lyrical and sonic themes hinge nostalgically on a longing for lazier times and may or may not involve references to inert romance or drug use.
I went on Hype Machine and Gorilla vs. Bear and downloaded every bloggable track by Neon Indian, Washed Out, and Memory Cassette -- I love chillwave.