A pet city is when you keep one AI city alive just so you can continue to play the game.
Now, the key civ is the civ that is linked to your science research. Depending on your power rating (supreme to pathetic) you are linked to a certain civ. If your rating is supreme then your key civ is purple (pathetic is white). So, the more techs you have above what the purple AI player has, the longer it will take you to research new techs on your own.
So, what you can do is beat the purple player (since they are usually your key civ) down to just one city (can be your pet city) and gift them all (or most) of your techs. Since they are small and you are being sure to subdue the city by surrounding it and such they can't really do much with the techs you give them. But, since you gift yor techs to them, they are equal (or near equal) to you tech wise, so the research costs for new techs you are going after is smaller than if your key civ was far behind you.
In GOTM 12 it saved me about 3 turns per tech just by making sure the purple player was close to my tech level.