This is a Derby 101 topic. If you're looking for more graduate level discussions, you might find this a little basic.
First, a 3-year-old horse is nominated by the owner or trainer for the Triple Crown. There is an early nomination period till January 25th when the fee is $600. Starting January 26th, the fee jumps to $6,000 till March 22nd. After March 22nd, if you have a horse who has done well in any Road to the Kentucky Derby races (more on those in a bit!), you can pay a supplemental nomination fee of only $200,000.
So if you have a horse that's 3-years-old and you think it might have a chance to run in the Derby, you really should get out your checkbook before January 25th.
As of 2013, Churchill Downs changed the process of how horses get into the Kentucky Derby. Previously, it was based on how much money the horse earned from graded stakes races. Graded stakes races are the races with the cream of the crop running in them. However, just because a horse earned enough money to get into the Derby doesn't mean it belongs in the Derby. The Kentucky Derby is 1 1/4 miles long. Some horses are sprinters and are better a shorter races. They might lead the pack for 6 furlongs or so but after that, they run out of gas and end up coming in 17th.
In 2013, they switched to a points system. There are 34 races from September through April where horses earn points on the Road to the Kentucky Derby. The first 18 races give 10, 4, 2 or 1 points to the first four finishers. Then near the end of February, it starts to get real and horses earn 50, 20, 10, or 5 points and then at the end of March you really get a sense of who will be in the gate on the first Saturday in May. Starting with the Florida Derby, horses earn 100, 40, 20, or 10 points. Keep track of who has how many points here.
In 2013, horses with anywhere from 14 to 150 points ran in the Derby. And don't disregard a horse with low points. Sure, Orb had 150 points and came in first but Golden Soul who entered the Derby with a mere 14 points came in second (and totally screwed up my bets!).