When police officers watch a suspect's house, keeping an eye on who's coming and going, they call it a stakeout. Most stakeouts are done stealthily, with the officers trying to avoid being seen.
Detectives on a stakeout often spend long hours sitting in an unmarked police car, watching a house or building. The stakeout might, for example, be used to prove that a suspect who claims to need a wheelchair can actually to walk, or that two people who say they've never met each other are actually close friends. The word comes from the stake that a surveyor uses to mark off a piece of land, and it's been around since the 1940's.