Phoebe's Cab initially belonged to her grandmother Frances. After the latter died, it was inherited by Phoebe.
The cab is first seen in "The One With Phoebe's Dad", where Phoebe drives it to upstate New York to see her dad, but she doesn't have the courage to walk to the house.
In "The One With The Bullies", Phoebe hits the dog of Frank Buffay's family with the cab.
It's also seen in "The One Without The Ski Trip" when Phoebe and her friends, excluding Ross, drive it up to Rachel's sister's ski house. They run out of gas and have to call Ross to the rescue.
In "The One With Joey's Big Break", Joey drives the cab to Las Vegas and discovers that Frances' ashes are kept in a box under one of the seats. Phoebe and Joey drive it back in "The One After Vegas", when Joey picks up a hitchhiker.
In the final episode, the Phoebe drives Ross to the airport with it. Ross also refers to it as "The Death Cab", based on Phoebe's driving skills. We also learned that the passenger's seat doesn't have a seat belt because "the paramedics had to cut through it".