The Constable is the villager who arrests Sims when the Monarch orders them to be sent to the stocks or into the pit. He appears usually out of nowhere and is large in size. The Constable escorts Sims to the stocks/pits by force with his giant sword.
When he comes to arrest a Sim (if the you are playing as that Sim), the player will get the message "Uh-oh, it looks like Sim's name is in trouble! The constable is coming to arrest him/her!". This message is displayed regardless if whether the Sim is marked for arrest or death.
It is impossible to marry the constable without the use of cheats. If you build up a relationship with him, the "Ask for hand in Marriage" interaction will be grayed out, saying that the Sim isn't interested. However, if the player uses cheats to marry him, he will randomly disappear to enforce the law. (It is possible to WooHoo and have a baby with him).