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How does “taking over payments” work?
The phrase doesn't mean anything specifically. Your SO could start paying the payments, but the title and lien would remain in your name.  If you wanted to change the title or lien to be in her name, you would have to sell the car to her (sales tax would be involved but the process would be relatively painless).  You could sell her the car for a pretty cheap price, but not $1. (unless the depreciated value of the car was less than the rest of the loan amount). You could draft up an agreement that if you break up or something, she agrees to buy the car from you for $x dollars minus all the payments she has made on the car.