It is unknown when Jesse acquired the Monte Carlo, but when the DEA came to bust the meth lab that he and Emilio Koyama set up, Jesse escaped in the pimped-out vehicle while Emilio was captured. It was then that Walter White saw the vanity license plate of the car, which read "THE CAPN", and thus figured out that Jesse was the meth dealer the DEA was searching for ("Pilot").
Later, Tuco Salamanca kidnapped Jesse and Walt and drove them out into the desert stuck within the Monte Carlo. Hank Schrader eventually located them using a Low Jack system Jesse had installed in the car. During the shootout between Tuco and Hank, the car's hydraulic pumps caused the car to "bounce." After the first car was destroyed in the shootout with Tuco, Jesse purchases the more practical 1986 Toyota Tercel from Badger's cousin Clovis.
The car was eventually bought by Rusty's TV and Movie Cars Museum.
There were actually 2 different cars, the 1982 with a hood ordainment and Grey interior only shown in season one because it was rented. They decided to keep Jesse on the show therefore they bought the 1982 Monte Carlo with a brown interior and without a hood ornament, it was featured on the show until it got shot up in the desert.
Jesse's Monte Carlo was selected during the Pilot (before Transportation Captain Dennis W. Milliken joined the show) and lasted until early Season Two.