The Aston Martin V12 Vanquish was supposed to make a special appearance in Twilight as the car Edward uses to take Bella to Prom. It got cut out in the book editing process however, and did not appear in New Moon or Eclipse at all, but did appear again in Breaking Dawn where it appeared as Edward's special occasion car. It was also the car that Jacob stole briefly from Edward when upset about the Bella triangle.
Bella never received the Aston Martin V12 as a gift from Edward. Bella's "after" car is actually a a Ferrari F430.
Bella gets it because Edward replaces her old car with the new bulletproof glass car. Bella didn't even know what the car was until random strangers came up and took pictures of it. She didn't think Edward was being serious about bulletproofing the car but he wanted to make sure she was safe.
Bella got the vanquish in Breaking Dawn when Edward replaced her car after it stopped working. Funny thing though Bella didn't know what the car was until a pair of strangers came up asking to take a picture of it and over heard them discussing the car's features. She thought it was a joke that Edward made that he'd get her a car that's bullet proof since she attacks so much danger.