This mitzvah includes both one who is paid to watch an animal (or other item) and one who pays the owner for temporary use of the animal. (For example, he might rent an animal to plow a field, pull a wagon, or as a means of transportation.) The Torah tells us that if the animal dies accidentally, the watchman must take an oath that it’s not his fault. If the animal is lost or stolen, the watchman must pay the owner for it. (After all, that’s exactly why the owner hired a watchman!) This case is more stringent that of an unpaid watchman, who was simply doing the owner a favor.
The Talmud in Shevuos 49a explains that there are four types of guardians: the unpaid watchman, the borrower, the paid watchman and the renter. However, there are only three categories of laws, as the case of the paid watchman and the renter have the same legal parameters when it comes to liability.
The many details of this mitzvah include the case of a watchman who subcontracts a third party to guard the item, the case of a craftsman who ruins an object he was given to work on, the renting of houses, the hiring of the animal’s owner to drive or work the animal, and more. (If one hires the animal and its owner, he is exempt from damage to the animal. And, for the record, there is a presumption that one’s adult family members will help guard an object in his care, but he may not sublet it to others.)
This mitzvah applies to the judges, who are obligated to try such cases. It is discussed in the Talmudic tractates of Baba Kama (chapters 6-7), Baba Metzia (chapters 3 and 6), and Shevuos (chapter 8). In the Shulchan Aruch, it is codified in Choshen Mishpat 303. It is #243 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos.