Every car on the road must be registered with a state. Registration connects the car to its owner and gives you license plates and a sticker that shows it is current. The details differ by state; see In your state.
The title proves who owns the car. When you buy, the seller signs it over to you. If you have a loan, the lender may keep the title until it is paid.
Registration is the permission to drive the car on public roads. It must be renewed, often every year or two. You get a card to keep in the car and a sticker for the plate or windshield.
A dealer usually does the paperwork and gives you temporary paper plates.
If you bought from a person, go to the DMV (or your state's title office) with: the signed title, a bill of sale, your ID, proof of insurance and proof of address.
Pay sales tax (in most states), the title fee and the registration fee.
Put the plates and sticker on the car.
Some states require:
Safety inspection: brakes, lights, tires, wipers, horn.
Emissions test (smog check): checks pollution from the exhaust. Often only in some counties or cities.
Other states require no inspections. Inspections are done at licensed garages or state stations. A failed car must be repaired and checked again.
When you move to a new state
New residents usually must register their car in the new state within a set number of days. Your old state's plates are not valid forever.
Forgetting the renewal date. Many states mail or email a reminder only to the address on file. Update your address every time you move.
Buying a car with a title in a different name from the seller's ID.
Driving with expired tags (stickers). Police can stop you for it.