It depends on the type of agreement you took. On hire purchase you make the final payment, pay any option-to-purchase fee, and the asset becomes yours. On a finance lease you usually continue with secondary rentals, sell the asset as the lender’s agent, or return it. On contract hire or an operating lease you hand the asset back.
Where the asset is returned, the lender will normally inspect it against fair wear and tear standards and against any mileage or usage limits in the agreement. Damage beyond fair wear and tear, or excess mileage, can result in end-of-contract charges. These are set out in the agreement, so check them at the start rather than at the end.
Where you keep the asset, you take on everything that comes with ownership from that point — maintenance, running costs and eventual disposal. Its resale value at that stage may be more or less than you expect.
Agreements do not always simply stop. Some roll into a secondary period or continue on a periodic basis unless you give notice, so diarise your end date and check the notice requirement well ahead. If you want to replace the asset, start the conversation a few months before the term ends rather than in the final week.