The picture is a screenshot of the phone app. You move to different physical locations gathering "exotic matter" or XM. The blue bar at the top is your amount of XM. The concentrated blue smoke puffs are "portals" of concentrated XM. When you are within about 50 meters, you can hack the portal getting portal keys, shields, XM bursters and resonators. Lots of other stuff plus media clips that tell the back-story to this game. If you find an unclaimed portal, you place "resonators" around the portal. You need eight. You need to place the resonators in strategic locations so that they are harder to destroy with the "XM bursters". Portals are usually public statues or artwork. Fountains, murals, or anything that in real life that attracts people. Once you claim three portals yourself or have hacked keys from the same team, you can create a "control grid" out of three. You level up in the game by getting "AP" basically experience points from creating the control grids. Depending on if you play for the Enlightened or the Resistance like myself, depends on what the control grids are used for.
Level 1 agents usually can only create control grids a few square miles but higher up levels such as Level 8 can span the country or from Europe to US.
I am in an area where like the player above was and had few portals to hack and take over. The good thing is that as player you can also introduce new portals into the game. I went to a couple places near me that had fountains, took a picture, and submitted to the in game app. A week later new unclaimed portals were now in the game. Yesterday I submitted five new portals and hope to have those claimed soon. I also took a trip downtown and hacked about 85 portals while riding passenger on a short trip.
Clan, militia, are all there too gathering members. The militia group I belong to is planning a raid in a few days utilizing the city's light rail and the concentrated portals that accompany the many portals along the route. Already over 200 members have pledged to go. If two or more people attack/defend the same portal, it helps as an accumulative effect. I am a three and my wife is a two. We can take down a level 5 portal together but separate it would take a few trips each. Large groups will speed this up and also generate plenty of AP experience points for all.