When countries such as Britain and France were at war, any ship belonging to one of the belligerents which was captured was known as a prize and the crew of the ship(s) involved in the capture were awarded prize money, based on the value of the captured ship. This value might be based on the number of guns on the ship if it was a Naval ship (a 'Man of War'), or value of the cargo, if it was a merchant ship or a privateer. If it was a British prize then the award would be made by a Prize Court which was part of the Court of Admiralty. Other nations has similar courts or adjudicators for this purpose. An officer on a ship which had captured a prize could expect a very large sum in prize money, and even ordinary seamen could earn as much as the equivalent of several year's pay in prize money. The system was generally phased out towards the end of the nineteenth century although vestiges of the custom still survive in the current day rules of
marine salvage.
As for your question about how a person from Guadeloupe might have joined the British Army, I assume this was a consequence of Britain capturing the island from the French in 1794 during the French Revolutionary War. If the man you think may have been your 3 x grandfather was taken prisoner during this raid, he would have been offered his freedom in exchange for joining the Army. This was beneficial in two ways: the British didn't have the expense of housing him in a hulk or prison and the prisoner gained his freedom and would be paid for his services. The Guadeloupeans were not keen on being ruled by the Revolutionary French Republic so he might have wanted to side with the British anyway.
