The three most important considerations when leasing a horse are: 1) finding a horse with a good temperament, 2) choosing an animal whose capabilities are compatible with your immediate goals, and 3) having a veterinarian check the horse thoroughly for soundness problems. I put temperament at the top of the list because a horse with a good attitude makes for safe and fun riding. The capabilities of the horse may not be up to your goals in years to come, but in a lease situation, it is fine if the animal is only suitable for your immediate goals. You can always lease another horse if your abilities surpass the current one. Soundness is also crucial, for a lame horse will waste your time and money.
The specific lease arrangements vary according to the worth of the horse. For example, if you are leasing a fairly old horse, then the owner my only require that you pay for insurance for the animal and absorb all of the fees attached to the upkeep of the horse, such as board, shoeing, and vet bills. If the horse is younger than ten years old or is a horse with a winning record, you may have to pay a lease fee as well, especially if you'll be showing in recognized horse shows or events.