The 4 buttons on top

Mail

When you click this button you will go to the mail screen, note that gray is its usual appearance, when you have a new message it will turn blue.

Economy

When you click this button you will go to hq screen, note that gray is its usual appearance, when you have a any new economy reports (buildings or units finished) the button will turn yellow.

Market

When you click this button you will also go to hq screen, note that gray is it's usual appearance, when you bought or sold something on the market it will turn green.

Military

As before this button will bring you to the hq screen, note that gray is it's usual appearance. This will light up red in 2 situations: if your fleet has reached it's destination and is awaiting order, or if you are under attack.