Free lease varies.. you pay nothing and you ride or you get the horse on your own property, do whatever but aren't responsible for any charges.
Half lease you rent the horse 50/50. So it gives you a more realistic idea of what the charges for keeping the horse are. A ball-park figure. Sometimes you pay half of the vet bill, half farrier, half food cost, half maintenance (like buying equipment, training costs, special tests, show fees etc.) It varies as well depending on the contract.
Partial lease is the ideal choice for someone that can't afford to invest in a horse entirely but still wants riding time. Partial leases range from 1-3 days a week average, you pay a smaller payment (not 50%!) of what it costs to keep the horse and whatnot. There are conditions and restrictions that are outlined in the contract.
PLEASE do not enter a lease without a contract!!!!!!!!! BAD idea!!