In many aviation jobs, the airplane requires a type rating (for airlines a type rating may not be required for FO's, but they still get almost the same training as a type). A company has two choices when hiring pilots...
1) Find a pilot who has a type rating.
2) Hire a pilot and train him.

Option one is preferred, but in many cases is not practical, so the company will incur $10K-40K in training expenses. A training contract is an agreement by the pilot to re-imburse some or all of the cost in the event the pilot leaves voluntarily within a certain time period. They are usually 1-2 years, the contract amount is usually $10-15K, and the amount is usually pro-rated (the longer you stay, the less you pay when you leave).

Note: Airlines will retrain all new-hire pilots, even if they already have a type-rating in the aircraft, so option one does not apply to airlines.

Not all companies have training contracts. A training contract means one of several things:

1) Companies that are so bad to work for that people leavea asap (mesa)
2) Corporate operators who provide a very marketable type rating. (CE-500)
3) Very small operators who would be economically damaged by one pilot leaving early.