Player Info Popups are exactly like they sound; click on a player's fleet of base and you will have the option to gain some basic info about that player. You will notice after clicking on them there is a "Player Info" option at the bottom of the screen just to the left-hand side of the normal info button:
As previously stated clicking this option will give you some basic info about the player in question:
Note that the info and options you get from clicking on a fleet or base are almost identical. Regardless of which one I have clicked on, I will still be shown the player's ID, base level, current planet and sector, total Medals, and the name of their current Alliance. I can also choose to ignore the player in chat from each pop-up as well at the bottom; if the player is already ignored then I will have the option to unignore them. Really the only difference between the two is the option on the bottom right-hand side. If i click on the player's base I can leave a bookmark there (more on that below), but if I click on their fleet I will be given the "Locate" option; clicking this will take me directly outside their base, so I can either then leave a bookmark or start moving fleets to them.