Those are "Pokéstops," or as we call them in human reality "places of interest." Basically, various notable landmarks in your general are get incorporated into the world of "Pokémon GO." In the case of my morning walk to the subway, for instance, a local church became a "Pokéstop," where I was given free "Pokéballs" just for tapping an icon and acknowledging the church's existence.
Beyond these local landmarks and running into the occasional wild "Pokémon," various areas near your location become "gyms." In the world of "Pokémon GO," these gyms are where you'll battle other trainers (you, the player, are a "trainer") with your captured Pokémon." Check out the killer kicks on my man right here.
In the (admittedly very limited) time I've spent with the game so far, it feels like a pretty incredible way to explore your town/city all while playing a video game of sorts.
There are some clear red flags, like the game not shouting "HEY YOU'RE ABOUT TO CROSS A STREET MAYBE LOOK UP FROM YOUR PHONE," but assuredly that stuff will shake out over time.
The other side of that coin is that the game is totally free.