Leases are very individual arrangements IME, so it's hard to give a lot of details. Generally, you'll find two classes of leases- partial or full.
Partial leases give you use of the horse only on certain days (with some being more strict about sticking to a schedule than others). Usually with these the horse will stay at the barn he's already in. Most of the ones I've seen the financial commitment is for a set amount each month and doesn't vary no matter what care the horse needs, though sometimes you'll split costs with the owner as they come up (for example, you'd pay a regular monthly fee for board, then when the horse sees the farrier you'd pay part of that cost to the owner)
In a full lease, you are the only one riding the horse, and you may be able to move it to your barn of choice (or home if you have space). You usually pay all regular costs of care, including board, farrier, vet, etc directly to the service provider. You might pay nothing at all directly to the owner (often called a "free" or "care" lease) or you might pay a small lease fee in addition to all the normal horsekeeping costs (especially if the horse is show-ready).
Some owners put more stipulations into their contracts than others. Before I bought my own horse, I did a few partial leases through the barn I was taking lessons at, and one full lease from a private owner. The partial leases through the barn were very loose with a short one-page contract. The barn arranged all care for the horse and I had access to their schooling tack, so I really could just show up and ride. If the horse was lame or had health issues, they'd let me ride a different horse of theirs. With the full care lease, the owner was a little bit pickier (understandably, since I moved her horse to my barn). Her contract stated some things she cared about- like she wanted the horse to be shod in aluminum shoes. It also specified who paid for vet bills in case of an injury/illness (in this case she carried a major medical insurance policy on him and I was responsible for the $250 deductible only) and allowed me to be released from the lease if he was unable to be ridden for 30 days or more.
IME, a clear understanding of who pays for all expenses (routine and emergency) and any clauses that allow you to terminate the lease early are important. You don't want to end up paying for a horse that you can't ride, and you certainly don't want to be the one to foot a $10k colic surgery bill! My full lease contract was a one year commitment, but had a trial written in; I could terminate the lease within the first month with no penalty. After that I was on the hook for the entire year unless (like I mentioned earlier) the horse was unable to be ridden due to injury/illness.
Overall, leasing was a very positive experience for me, though I was certainly ready to buy my own horse after a few years of it!
“The horse is a mirror to your soul. Sometimes you might not like what you see. Sometimes you will.” - Buck Brannaman
"Nothing forced can ever be beautiful." - Xenophon