Cleaning is the process of removing material from the surface.
Sanitizing is the process of reducing the number of organisms (in brewing, we're worried about bad bacteria, mainly - but others also like fungi and unwanted yeast).
Sterilizing is like sanitizing, but removing ALL microorganisms (any living being - microscopic).
If I remember right, sanitizing is a technical term that means a certain allowable amount of microorganisms remain, and sterilization is removing ALL of them.
For the most part, brewers don't need to sterilize, only sanitize. The chemicals made for brewing are made to sanitize.
You need to both clean AND sanitize. The analogy I like is this: if my Dog pees on my floor, I first wipe up the liquid (clean), then use bleach (sanitize). If I merely use the bleach I've still got pee on my floor.