Influence is a type of cultural control that is generated by every planet. The amount of influence generated depends on the population, racial bonuses, and certain projects (like embassies). Influence determines the extent of the color coded borders on the main map. Influence also affects tourism, with the amount of tourism being proportional to how much influence you have. Influence can also cause colonies to rebel, or return colonies on the verge of rebelling. If a colony is in an area of cultural influence that is four times as powerful (or greater) than the amount the colony is generating, there is a chance that the colony will rebel. A pirate flag on a colony indicates that it is in danger of rebelling (and steps should be taken to increase influence from that colony if it is to remain). Influence generated by your planets is also stored for use in the United Planets votes each December. These influence points may be traded, allowing civilizations to either cash in on or purchase some additional voting leverage.
On the galaxy main map, selecting a planet will reveal that planet's influence level (shown as IP points). In parentheses next to your IP level is the relative foreign influence calculation. If that number is greater than 4, your planet may culturally revolt. If your espionage level is rated "Low" or better, you can see whether or not another civilization's planet is close to defection.