Bata bata (ばたばた) means rushing around: busy, noisy and in a hurry.
The alarm didn't go off. You run from the bathroom to the kitchen to the door, looking for your keys, your bag, your other sock. That morning is ばたばた.
What it means
ばたばた started as a sound: running feet on a wooden floor, or a bird flapping its wings. From there it came to mean being very busy, with no time to stop.
How to use it
With します it means being busy: ばたばた します, ばたばた して います. Right before a verb, it is the noise: ばたばた はしります. People also use it politely, to say sorry for being slow: ばたばた して いて、すみません.
ゆきさん、はやく！ でんしゃが きます！
Yuki, hurry! The train's coming!
まって ください！ かぎが ありません！
Wait! I can't find my keys!
まいあさ ばたばたですね…。
Every morning's a rush with you...
Close cousins
どたばた is noisier and messier: chaos, slapstick, everyone running into each other. A どたばたコメディ is a slapstick comedy. のんびり is the opposite of ばたばた: slow, easy and relaxed.
In manga and anime
A late character runs with toast in their mouth, legs drawn as a blur, and バタバタ in the dust behind them. Birds and flags flap with バタバタ too.
Can you use it?
Yes, anywhere, even at work. ばたばた して いて、すみません is a polite, everyday way to say “sorry, things have been hectic”.