Manifest Destiny: faster tile expansion or perhaps bonuses related with the adoption of democratic social policies.
Trade Caravans: probably advantages towards creating trade routes, perhaps tightly related to a new economic system or mechanic yet to be introduced
Sacrificial Captives: Bigger bonuses to destroying cities: you gain extra culture and / or gold per citizen killed.
Art of War: Spy bonuses, probably an additional line of vision into the enemy territory?
Monument Builders: Bonust to the production of wonders, obviously.
Sun Never Sets: Less penalties for having cities in far away places.
Ancien Regime: Some kind of culture bonus for adopting certain social policies like aristocracy.
Hellenic League: Doubles the bonuses provided by the alliances with other civs or civ states
Population Growth: Duh.
Bushido: Special unit promotion unique to Japanese units, only applicable to melee - based units (swordsman, spearman, etc).
Barbary Corsairs: Extensive bonuses for disrupting enemy trade routes.
Archaemenid Legacy: Culture bonus, probably related to stability of goverment and the manteinance of traditions
Siberian Riches: Trade and food bonus in tundra and artic regions, or perhaps extended bonus of resources: say, if usually one oil resurce can support 5 tanks, with siberian riches it can support up to 10 tanks.
Father Governs Children: Bonuses for choosing "paternalistic" social policies (monarchy, social democracy, etc).
River Warlord: Units gaining attack and defense in the proximity of river tiles.