Each player in the game has an icon across the top of the screen. Clicking on the icon once will center the map on the player's home star. Clicking twice will show the player's Empire Summary Screen.
The Empire Screen provides many clues about how well the player is doing in game. You can see the total amount of Economy, Industry and Science they have on their stars. This tells you how much money they receive, how many ships they are building, and how fast they are unlocking new technologies.
The Empire Screen also shows how advanced the player is in each of the games Technologies. You can see how effective they will be in combat or how far they can see in the map for example.