Color correction is used to make the colors more accurate by fixing white balance, black levels, saturation, contrast, etc. It is also used to match shots and elements in compositing. Color correction can even go into stabilizing flicker problems in your footage. Always color correct your footage before color grading your footage.
Color grading is used to give style and an artistic look to footage and set the mood or tone. Get great tools like Blackmagic Design DaVinci Resolve Studio, Red Giant Magic Bullet Suite, Pomfort LiveGrade, LookLabs, Boris Continuum, Synthetic Aperture, Koji, and more!