Why should I "watch" a nearby Pokémon?

"Watching" a Pokemon allows you to get the distance to it (measured in footprints) on the main screen. It's purely a time-saver to keep you from having to constantly open/close the Nearby screen.

Additionally, Watching a Pokemon will cause the nearby screen to have a green "blink" when you're headed in the Pokemon's current direction. You can also do this without selecting a watch, as Pokemon will move around the menu, with the nearest always being in the top left.

There are two things to look out for when hunting Pokemon.

1) The number of feet - the number of footprints drops from 3-2-1-0 the closer you are to the Pokemon.

2)The Pokemons position on the nearby pokemon list (or Poketracker as i refer to it). Top left is closest to you, bottom right is farthest.

When I am hunting a specific Pokemon I keep the Poketracker up , if the silhouette moves left I know I am getting closer and if it moves right I know i am getting further away. Think of it like a very sophisticated version of marco polo or Hot&Cold.

Most of the answers here are outdated because the system has changed since they were posted.

The Nearby pokemon are only the ones that are currently near pokestops that are near you. If there aren't any stops it will simply show the pokemon that are somewhere near you and is of little use. Although then you can use it as in the older configuration; disappearing from nearby means either that the pokemon is in the opposite direction and is too far to be visible or it has just disappeared.

When you want to catch a pokemon that is nearby and by a pokestop, by clicking that pokestop it will show you on the map where related to you that pokemon is and you can more easily get there. Also it will start tracking that pokemon and will notify you if that pokemon disappears before you get there so that you don't go there in vain. Although some times you might get notified that that pokemon isn't there anymore because of bad connection or when you are already quite close to that stop (from personal experience).