What is AT&T's signal booster called?

That depends on if you are asking about a booster or microcell.  The microcell is not a booster.  It provides signal via your cable home internet connection.  It requires wired home internet and a postpaid plan.  

If you are looking for an actual booster, that is a device that requires electricity to work.  It amplifies an existing signal.  

I have both.  A microcell at home and a booster for my car.

https://www.att.com/att/microcell/

If you want a booster, the Weboost website is where you want to look.