Culture, also referred to as Influence, represents how much your own people and other people respect your civilization. This is separate from "military respect".
Each planet generates a certain amount of culture/influence. Racial abilities, certain wonders, and influence resources affect your Culture Ability.
If your influence is more than 4x that of the native alien influence on a planet you don't control, there is a per-turn chance that this planet will culturally defect and decide to join your civilization. This is indicated by the number in parenthesis next to the influence number.
If your influence is more than 4x less than alien influence on a planet you control, there is a chance that the planet will defect away from your civilization. The ratio of alien influence to your influence is again the number in parenthesis next to the influence number.
A planet ready to defect is indicated by the skull-and-crossbones icon.
Minor races appear to be immune to cultural conversion.