I'm not sure this question falls into "Motor Vehicle Maintenance & Repair" but generally the procedures are similar across states with some variations.
In California there is an official document call a TITLE that shows ownership and has a place for the seller to assign ownership to the buyer. Generally when you purchase from a dealer, they are, for the most part, bound by state laws governing their conduct. With a private seller, well, all bets are off. BE CAREFUL! (You should also be careful and check reviews and reference for dealers.)
Things to be careful of include:
- Forged titles or titles in another person's name from the supposed seller.
- The VIN number on the car being different from the vehicle being purchased.
- SALVAGE or REBUILT title status. This means the vehicle was "totaled" and someone did work to restore it to drivable status. They may or may not have done good work.
- Liens, the title may have a lien on it which means someone used it as collateral for a loan. Be sure any liens are removed before you hand over any money.
- Don't EVER send money to someone on a promise. There are so many scams here that I don't have room to list them all but be aware that private selling of cars is a prime scam target. Again, BE CAREFUL. If ANYTHING is not up-and-up RUN AWAY!
- Be very careful meeting for an exchange. It's not unheard of to be robbed since you have money and they know it. Meet at a police station or a bank or someplace where you will not be vulnerable.
There are a lot of other things to keep in mind and I'm sure some others will jump in and list them too.
Once you have the endorsed title you will take that to the DMV and apply for a new title in your name. Plan to spend some money on that and I think CA also will collect sales tax on your purchase. You'll also need to get a registration and license plates at the DMV. That, of course, costs more money. To get that you'll need proof of insurance and I believe a CA drivers license as well.