Atomizer - this is basically just a cup with a coil and couple of wicks and/or a bridge inside it. Dropping liquid onto the coil directly is how most of these work, but all systems have this basic simple concept, coil, little bit of wick, and some liquid to produce vapor. A direct atomizer produces the best vapor, flavor, and throat hit of any attachment, just most attys dropping 3-4 drops of liquid into it each time need a vape can be a little tedious, I mostly drip/atty at home.
Cartomizer - this is an upgrade to the straight atty that I have seen, little longer barrel, atomizer coil is generally vertical compared to horizontal like in a atty above, with some type of filler surrounding the coil and an air tube. These the filler soaks in the liquid and feeds it to the coil, thus becoming the wick and replacing it to an extent. These you can fill more liquid into and are nice for portability without much leaking and less time between refilling.
Tanks - Come in different forms, cartomizer tank (DCT), clearomizer, atty tank system like the AMP tank. Tank all it does is hold extra liquid. Like cartomizers and hate refilling them or running out of juice very often for the carto, get a carto-tank, punch or slot your carto, prime the carto with liquid like normal first fill, insert into carto tank, fill the carto-tank up, slide fully seated, put on tip, vape away, the liquid in the tank feeds the carto. Clearomizers, some type of wick feeds liquid to the coil from the tank, larger tank, longer between refills. Atty-tanks are a hybrid of a DCT and clearo, having the advantages of both a tank and a straight atty, same flavor and such of an atty compared to other attachments, just don't have to do a direct drip al the time.