I'm most familiar with the War of 1812, so that's what I'll elaborate with.

Captured vessels were usually sent to a home port of the nation doing the capturing with a prize crew (some of your original crew aboard your ship. You would haul the ship to port (in this war, usually Halifax) and a judge would hear the case whether your prize was lawful or whether you were illegally seizing ships or breaking the blockade rules. If you were found correct, the ship and cargo would be auctioned off, and if not it would be returned to its rightful owners. And none of this was immediate, it was a lengthy and expensive process.

How much you got paid was revised in 1808, but during the war in question the breakdown for pay is as follows

Captain (2/3)(2/8)

Naval lieutenants, masters, physicians, captains of marines (1/8)

Midshipmen, senior petty officers (including gunners, bosuns, carpenters) (1/8)

The remaining half was distributed to petty officers and ratings, with higher ranks dominating the amount of shares received. So every person on the ship had a stake in capturing a prize. And the the 1/3 of your take as a captain goes to your admiral. For example, Rear Admiral Cockburn netted some 400,000 pounds during tenure of command during the War of 1812 despite only being in command a brief period of time.

Sometimes ships paid a ransom (often illegally) to avoid capture and were sent on their way, but these vessels ran the risk of being captured again. Nearly a third of all vessels captured by the United States were actually recaptured by Britain on their way back to the United States.

Between June 1812 and March 1815 nearly 500 ships arrived in Halifax as prizes, but getting your money was no easy task. Though blockade could be extremely lucrative it often took from six months to over a year a year to get your prize money (something that caused a lot of anguish within sailors ranks and helped fuel desertion). It often took even longer if there were multiple ships involved in the capture and everything had to be adjudicated. In fact, many disputes weren't even settled until well after the war, and many of the people in question could no longer be contacted and paid out.

Sailors weren't the only ones who could receive prize money though. Major General Isaac Brock netted him some $60,000 during his capture of Detroit (which was fortunate because his family had some large and outstanding debts). A small raid by Lieutenant Colonel Joseph Morrison on Madrid after the Battle of Crysler's Farm in 1813 collected many public goods and hauled most of them back to Canada (which were public goods held by the American government to be put up for auction) and netted each private involved in the endeavor about 2 weeks pay.

How Britain Won the War of 1812: The Royal Navy's Blockade of the United States- Brian Arthur

Field of Glory: The Battle of Crysler's Farm 1813- Donald E Graves