PCP:
Suppose you can afford a £10k car but you fancy a £20k car (I have no idea of the actual numbers in any of this, this is just vague ramblings). Then PCP the dearer car for 3 years. After 3 years, you pay the per-arranged balloon payment and own the car, or hand the car back, or swap for another car (along with some more money).
If you can afford the £20k car without PCP, you may still want to PCP as you can use the money that you would use as a balloon payment for other purposes (either investment, either keeping it, whatever). After 3 years, once again you can choose what to do with the car (buy it, swap it, give it back). PCP also gives the advantage of (if you don't want to sell the car privately) just handing the car back. You won't get as much money as selling it privately but you also won't get the hassle of selling it privately (you may like/dislike selling privately).
PCP also has limited mileage contracts and it's usually, IMHO, a good idea to go for a deal where you have a higher mileage contract than you the actual miles you drive.
Maintenance: some car companies have exorbitant service prices and you are obliged to service the car during the PCP. To alleviate this high cost, the companies like to sell maintenance plans. You can also buy tyre insurance, wheel trim insurance, ding insurance, things like that.
You can usually get a better deal from a PCP finance company rather than a dealer. There are a few terms you need to be familiar with (I cant recall the names, someone might chime in). Loads of people PCP these days and it's an option to keep up with the Joneses.
Leasing:
IDK much about PCP, and know even less about leasing, except that with PCP, you have the option of owning the car at the end, but with leasing, you never own it.