Hire Purchase is a straightforward finance agreement leading to ownership once all the payments have been made. The amount of each repayment and the length of the agreement are fixed in advance.
The interest rate and agreement length are fixed at the start. Initial payments can be low on hire purchase, or you can choose a higher deposit, including using a part-exchange to cover this. You can pay the finance off in equal monthly instalments, or you can add a final "balloon" payment, which will reduce your monthly payments by delaying paying off the last chunk of the finance until the very end of the contract. Unlike a PCP contract, this balloon payment is not optional, so you will need to ensure you can afford to pay it when it becomes due.
Once you have made the final payment, the vehicle is yours.
Lease purchase is a term used to describe a hire purchase with a balloon payment (in which case hire purchase normally refers to an agreement without a balloon). Other than the balloon payment, there is no difference between hire purchase and lease purchase.
As the name suggests, with hire purchase you hire the vehicle during the agreement. At the end, you pay an option to purchase fee, which is required to legally transfer ownership from the finance company to you. A conditional sale agreement works in an almost identical way, except that there is no option to purchase fee — ownership passes automatically to you once the final payment has been made, without the legal necessity of a nominal fee.