Leasing a horse means renting a horse for use. You can either rent it yourself or with your friends. When leasing there are some fees involved which you need to agree with the owner or the person renting out the horse.
Leasing a horse pretty much means "renting" it from the actual owner. There are a few different kinds of leasing you can do. A full lease is when the owner let's you keep the horse either at her barn or where ever and you must take responsibilities for all the fees and costs. Half lease is when owner pays part of the fees and so do you. Then there is on farm or off farm but that is self explanatory.