Matt:
If you enter a horse in a Claiming race, basically you are putting him up for sale for the specified claiming price. If you own a 20K claimer, you can enter him in a 30k event. Chances are noone will steal him from you, but he obviously has a lesser chance of winning the race. Put him in a 10k event and he'll get you purse money...but will probably get taken from you by a bargain hunter. This idea is meant to keep the game honest and competitive.
To run in an allowance race means you usually have to meet the specified conditions of that race. The horses are not for sale, and are typically faster than the claimers. An example of a conditional allowance would be "non winners of 2 races since July 21st". Self explanatory.
Maiden races are for nonwinners. Every horse in the race has yet to win, or "break his maiden". Using the same logic above, there are Maiden Claimers and Maiden Special Weights. MSW are for the faster horses, usually better breeds. Entering your horse in a MSW ensures you again that no one can claim him...but he'd better be fast or he'll be a maiden a long time. Competition in MSW is tough, especially at the premier meets.
Hope this helps.