It may sound strange and unnecessary, but I want to use this well-known word in my country in an international thing. That is why I need your knowledge/definition of it if there's a one.
It could mean anything: a cartoon character's name, a plant, a meal... etc
So, please share your thoughts about the word "Babai".
Another thing: I'm not asking English speakers only.
In the Russian-speaking and Ukrainian-speaking environment (maybe just on the territory of the former USSR), this word means exactly what Pavel said: "Its a made up creature to scare kids if they behave bad".
I also came across its use in the sense of: the impolite name of the representative of the Central Asian republics of the USSR (Uzbekistan, Turkmenistan, Kyrgyzstan, etc.). Apparently, in their speech, incomprehensible to Russians, the word Babai was often encountered, since in their languages it is father or grandfather. Why "impolite name"?.. Well, since their speech is not understandable, they do not know how to speak Russian, and, accordingly, they are not worthy of respect. It was a USSR-reality...
Babai in Farsi (Persian) means Dad.
It is a friendly way to call your father.
Babai ( Dad ) --> Pedar ( Father )