A 3 way is a combination of a woofer and an added tweater, this is inorder to cover more of the entire sound spectrum. A 3 way can usually handle low bass (no where near as low as a sub), and handle mids (cuase it is a woofer), and then it can handle the upper highs cause of the tweet.
A 2 way is a 3 way minus the tweet. So basically it will handle the lows and mids, down do a certain freq. and up to a certain freq. so it only covers a certain band of freqs.
What to go with all depends on your setup and preference. Some say go with the 3 ways always and just add crossovers to them, others say go with 2 ways, and add some tweets. It is all up to what ever you want to do.