Through the Ages your people will grow to crave the Luxuries of the world, as evedent by being repeatedly assigned tasks to capture various luxuries. Luxuries, more commonly known as Luxury tiles or simply "lux tiles" amoung players, are secured capturing them with your armies on the World Map. The positive effects of Luxury tiles can be enhanced by various Research topics and Buildings.
Once a luxury is captured it will contribute slightly to the amount of Culture gained by your empire per hour; capturing many luxuries will significantly speed you along to your next wonder, while a single luxury will hardly register on the culture meter.
The other major effect that holding luxury tiles has on you empire is an increase in the mood of the settlment which captured the luxury. The mood boost effects translates only to the settlment that sent out the army that captured the luxury. For example, if you capture a luxury with an army that marched from one of your settlments only that settlment will get a mood boost, not your capital or any other settlment.
The mood boost is extremely important to consider, especially in the feudal age and beyond when money is scarse, meaning high taxation and unhappy populations. Because the mood of your people directly effects building and research timers as well as the speed at which resources are gathered, capturing luxurys can actually directly speed up your empires progress.
Luxurys are critical for balancing out the mood of a settlment, allowing buildings such as Lodgeing to be constructed without worrying about angering your population. Also, luxuries allow high tax rates to be maintained without angering the population.
Luxuries available upon reaching the Classical Age
Luxuries available upon reaching the Renaissance Age
Luxuries available upon reaching the Modern Age