A protectorate is a state that is controlled and protected by another sovereign state. It is a dependent territory that has been granted local autonomy over most internal affairs while still recognizing the suzerainty of a more powerful sovereign state without being its direct possession.

What is becoming protectorate?

When a minor nation has suffered great military defeats and/or lost a considerable amount of land, it may offer to become a protectorate rather than face complete destruction.

How do I make someone protectorate in Rome Total War?

The best option in this situation is to first go and ask for a protectorate with a decent diplomat. If they refuse then attack their richest and largest provinces and exterminate them. Once they are down to this one province ask them again. They should be more that willing to accept a protectorate off of you now.