Nearby sits in the "sightings" section at the bottom right-hand corner of your display. This has always shown Pokemon that are near you, or thought to be near you, but it's never really been very useful. Now it is.
Tap the bottom right-hand tab to open up the Nearby screen.
Pokemon nearby are shown with a picture, taken from the local Poke Stop.
Tap on a Pokemon you want, then tap on the footprints below the location and Pokemon picture.
The map then zooms to a top-down view, showing the area, near a Poke Stop, that the Pokemon was sighted.
When returning to your normal Pokemon Go navigation view, that selectedPokemon is indicated with a pink footprints indicator, so you can walk that way and catch it.
When you arrive in that location, you'll either find that Pokemon there, or you'll get a message to tell you that it has fled the scene.
The good thing about the Nearby feature in Pokemon Go is that it helps you make decisions about which way to go and should make it easier to locate those Pokemon that you're yet to catch.