When you have a palace at level 10, 15 or 20 or a residence at level 10 or 20 you can, if you have enough culture points, found a new village or conquer one.
To found one you just need three settlers and an empty spot on the map. To conquer a village you need a Senator/Chief/Chieftain and an already existing village on the map.

How do I found a new village?

In order to found a new village you need 3 settlers, an empty spot on the map and enough culture points. You do not have to research settlers but can directly build a set of 3 settlers with a level 10, 15 and 20 palace. With a residence you can only train 6 settlers in total, 3 at level 10 and 3 at level 20. To build the palace you need a main building level 5 and an embassy, for the residence just a main building level 5. After training the 3 settlers you can choose an empty spot on the map and click "Found new village" (Hint: you need 750 units of each resource!) then your settlers will leave their home village and found the new village for you.

How do I conquer a village?

See conquering villages for further reference.