It really depends on the individual situation, your location, what the quality of the horse is, etc etc.
Around here you can find "back yard" trail horses for $100 a month for unlimited rides and that includes everything (ie: you don't pay anything extra like vet/farrier/feed). Or, you might chose to lease a horse you can compete higher levels with and pay one lump sum for the season ranging anywhere from $5000-$50,000 (sometimes more or less depending). And then you can find anything in between.
Part leases - generally this means you ride 2-4 days per week (one should be a lesson day which is not included in lease price). You pay a monthly flat rate which is a portion of the horse's board, but can include a portion of vet/farrier/feed costs as well. So this could range anywhere from $150-$500 in my area - again depending on quality of horse and how much that particular horse's board is/type of barn.
Full lease - means you are the horse's primary rider and ride 5-6 days a week, again one should be a lesson day that is not included in your lease fee. Lease fee generally includes same things as part lease, only in this case you'd of course pay a larger portion of board (possibly all of board) and depending on situation could also include vet/farrier/feed bills in that same flat rate. In my area full leases can range from $400-$900. Again, this will not usually include your lesson fees.
Free off property lease - you don't pay the owner of the horse anything, but you do pay everything else. So that means you can take the horse off the owner's property to your own barn and you pay the board/vet/farrier bills. Generally for a free off property lease you also need to supply all of your own stuff (tack, blankets, etc) for the horse as well.
One of my mare's has been free leased out in the past. Usually for 1 full show season. In my situation, her leaser is responsible for all costs, as well as all tack/misc. items. They however do need to keep her in tack that I approve of, including no use of gadgets and must stay in her current bit/bridle combo. They are responsible for routine vet/chiro/farrier care. If there is a non-routine vet bill that exceeds X amount, I pay it as I am the owner and that's just what I am comfortable doing. Occasionally I will send her blankets on lease with her if absolutely necessary as she is size 86" in most brands and they are not the easiest to find around here.
So point being, leases really vary from area to area, as well as type of horse/use of horse, etc etc.