Types of leases (and cost) depends on area.
What others are describing as a "full lease" (taking over horse cost) around here would actually be considered a "free lease" where leaser pays for horse's care (but no lease fee).
The terms of a lease will vary greatly from area to area, so do your research locally. And even if an owner describes it as one thing, their specific terms may vary as well.
Free lease - Not actually free, only free in the sense that you do not pay the owner. Typically not on the owner's property. You move the horse to an approved facility off property. You pay the board there, as well as vet/farrier/feed. You supply everything (ie: tack, blankets, whatever else you need to supply). However, even though you supply all these items, the owner is still in charge of what you can and cannot use on said horse. For example, I have leased one of my mares out as a free lease (as well as paid free lease) - I had in contract that she may only be ridden in X bit, no martingale, regular bridle, etc etc. Leaser needed to supply those things for my horse.
Paid free lease (sometimes just called a paid lease) - Exactly the same as free lease, however the leaser also pays a lump sum to the owner. Usually for the show season. Aside from that, everything is generally the same.
Full lease - Generally takes place on the owner's property (or wherever they board). Horse generally doesn't move facilities, unless going to a show. Leaser pays a lease price every month. That cost goes toward the horses board. Owner is still responsible for everything else. Sometimes owner provides tack, sometimes the leaser. The cost is generally close to, or equal to, the cost of the horse's board. So, if board is $900, lease is generally $900 or there abouts. Leaser rides 5-6 days per week.
Part lease - Same as full lease, except it is part. Leaser pays part of the horse's board. Leaser rides 2-4 days per week on average. So if board was $900, part would be $450, give or take depending on if only riding 2 days, vs. 4 days.
Keep in mind that that is what leasing entails where I am. Things can vary from area to area, so best to do you research locally.