If you buy a car on loan, the lender gets title, and the title is registered with your state government. The lender gets title to your plane, too, except the registrar is the Federal Aviation Administration.
When you buy a plane you’ll fill out an aircraft registration application with the name(s) of the owner(s) of the plane, the registration number (for certified aircraft it’s the N-number), make, model, serial number, and relevant signatures, and, of course, you pay a registration fee. You’ll also need a bill of sale signed off by the seller and any lien holders.
Make sure the plane you buy is available for sale. That is, the possessor of the plane might not be its legal owner. Title might be held by a lender, a corporation, a partner, or a relative of the “owner.” You can check with the FAA to find out who the registered owner is. If you’re using a lender, the lender will do a title search to make sure the title is clear of any problems that would preclude transfer to the lender. Both EAA and AOPA also have aircraft title-search services.
"Man cannot discover new oceans unless he has the courage to lose sight of the shore."
-- Andre Gide