I'm assuming you're talking about leasing a horse? First Time Leaser: Tips, Suggestions and Must Haves!
Check out that thread, the question is somewhat the same as yours.
You can also do lease to buy options where you pay a set monthly fee which accumulation into the total price. Of course then you'll have to check liability with what happens if the horse loses value due to injury or bad training while in your care, etc.
Leasing is a great opportunity, and I would go for it. Basically you and the owner need to work out what fees you cover, but as in the thread above, it is generally assumed that you cover board/vet etc, unless special circumstances. For most lease situations it's exactly like owning a horse, only you can "give it back" if it doesn't work out, or you don't have a lifetime commitment. The most important thing is to be clear and concise with the owner and to talk about everything beforehand. I hope this helps, and if you have any more specific questions, feel free to ask =)