“Leasing” a car is not something you’d do for a summer.
A car lease is basically a financing option. For all intents and purposes, you are buying the car. It is registered in your name, you pay the registration and insurance, you are obligated to maintain it, if it gets stolen it’s your responsibility.
New car leases generally run between 36 and 60 months. The lease payment is a function of (i) the initial price of the car, (ii) the estimated value of the car at the end of the lease period, and (iii) the finance rate (interest factor). At the end of the period, you can either give the car back (and they charge you if you’ve driven too many miles, or if there’s any damage to the car, as that affects the value in (ii), above), or buy it for the estimated value (the residual).
Lease payments are less than loan payments, because you’re really only paying off the difference between the initial cost and the residual, plus finance and carry costs.
Up front, you pay license fees and registration, plus first month’s lease payment and taxes. You may also pay a “capital cost reduction”, which is basically an upfront payment that will reduce the amount of the payments over the term.
You negotiate a lease just like any other car purchase - the amount of the lease payments depends on the amount the dealer charges you for the car, so the lower you can get that number, the lower the lease payments. The residual and interest factors are generally set by the finance company, and aren’t negotiable.
At the end of the term, if there’s no damage, and your under the milage limits, you can give the car back for free (even if the initial estimate of the residual value was wrong).
“Renting” a car is what you’d do for the summer. I’m not sure where you’d go about doing that. Craig’s list?