You have to take at least 3 cities of a foreign culture to get the culture. I don't honestly remember what all the abbreviations mean anymore, they made it really complicated now.
Depends on the map size, e.g. the Giant map requires 6 cities of each foreign culture. As far as I can tell/remember, it works like this:
C.N. = Culture Native, which you start with
C.L. = Culture Local, which is either built manually or automatically in each city depending on how you get it. If you found a city, you automatically get the C.Ls. that corresponds to your C.Acs. If you capture an enemy city, it gives you the corresponding foreign C.L. (e.g. if you're playing as China and capture a Greek city, you get C.L. European in the captured city), but you need to build your other C.Acs. there. However, if one AI captures another AI's city, and then you capture that same city, it doesn't seem to ever have any C.Ls. inside (e.g. if Greece captures an Iroquois city, and then you capture that newly-Greek city, you get neither C.L. European nor C.L. North American).
C.Ad. = Culture Adopted: If you get at least 3 cities of the same foreign C.L. (depending on map size), your capital automatically builds this, which makes all of your cities able to build the relevant C.L. and gives that C.L. to new cities that you found.
C.Ac. = Culture Acsomething: All of the cities you found automatically get the corresponding C.Ac. for their C.N and their C.Ads.
If you want to build cultures, pay attention to C.L., since that's the one you need in each city.
