I am making a control circuit for charging of a 12 V battery, and I'm having problems starting because I can't figure out how I could measure that the battery is fully charged.

If I apply 12 V (DC) to the battery, and the battery becomes fully charged... the node with the applied 12V and the battery 12V will be the same... or?
So how can I know?

And also, if applied 14 V for charging the 12V battery... what would happen?

 

Try the big battery resource - http://www.batteryuniversity.com/

 

A 12V battery is not 12V when it is fully charged.
It will barely charge from a voltage as low as 12V.

 

Just saying "12V battery" is far too generic.
Lead-acid, sealed lead-acid, NiMH, Ni-Cad, etc. etc... there are many different basic battery chemistries, and then there are different construction methods used within the various categories, and a wide variety of Ah or mAh ratings in each type.

They all have different charge /maintenance requirements.