Yep, what Tinyliny said. It can vary from paying a small fee to ride a horse one day a week (maybe even only in lessons, but at least you know you'll always be riding the same horse), to paying the full costs of owning a horse, and having full access to the horse without actually paying the up-front cost to purchase a horse (or having the full benefits of legally owning the horse, either, of course), to paying a monthly (or yearly?) fee MORE than the costs of that horse's upkeep, for the benefit of using a horse with really good training and/or show records. There are even breeding leases, where the person leasing the horse doesn't ride it at all, but is just leasing the mare for the purpose of ending up with a baby out of that mare, and returns the mare but keeps the baby after all is said and done.
Just about any sort of agreement can be written up and called a lease, as long as both parties agree. I think partial leases and full care (free) leases are common ways for people to bridge the transition between taking one lesson a week, potentially on a different horse each week, and full-on ownership. Partial leases give someone the ability to at least ride the same horse every time, and possibly a time or two each week outside of lessons, and full leases give almost every benefit of horse ownership, without the up front cost or full commitment (i.e. if that horse ends up not being a good fit, you can return it to the owner easier than it can be to sell a horse). And sometimes the full-time leases are more of a "lease to buy" option, if either or both parties are potentially interested in selling/purchasing the horse, but want to give it an extended trial period first.