A proposition player, usually referred to by the shorthand word prop, is an employee of a casino whose job is to play in poker games as directed by the floorman, usually in order to keep games going as they begin to get shorthanded. The casino has an interest in keeping more games going, since they earn their money by raking the pot: the more tables in play, the more pots, and the more rake the casino can earn.
A prop is usually paid an hourly wage (same as most employees of a casino) and must gamble with their own money as they play the games to which they are assigned. A prop generally has no say over which game to play (the floorman assigns them to whichever table most needs an extra player), and usually has no say even over which variant or stakes to which they may be assigned, though some casinos may offer ranges of props (e.g. low stakes or high stakes props) so that a prop isn't trapped having to play $4000/$8000 limit if they only have a $20,000 bankroll. A successful prop must be willing to play in any game at any time for whatever period of time the casino desires.
Proposition players are required to identifiy themselves as employees of the casino when they sit at a table. In most states, this requires the prop to wear an ID badge during play that identifies them as a proposition player to anyone who looks.
Props may be offered other amenities by the casino, such as free food or drink, free rooms, etc, but in general they are not offered extra money with which to gamble - they are merely an employee earning a wage like any other employee of the casino. They are required to act with some level of decorum, but are not required to perform other casino duties (such as acting as floor or security) and can get into rules arguments just like any other player: at the end of the day, though, their daily paycheck is paid by the casino, and any prop who makes players unhappy or drives them away from the casino is unlikely to be employed for long.