Allowance Race
A race with non-claiming conditions set by the racing secretary. They are meant to be a step down from stakes racing.
Rabbit
Also known as a pacemaker; this is a horse with no expected chance of winning but is used to prompt a quick pace on the front end of the race to the benefit of a stablemate who is a closer.
Handicap
A race where a handicapper has assigned weights to be carried by all runners on the basis of their past racing efforts.