I heard it called "the base" before also. I was curious, so I looked it up online:

Bin Laden himself said in 2001, "We used to call the training camp al-Qaeda ["the base"]. And the name stayed."

So I guess that's where "the school" definition derived from, a "training" camp.

Otherwise Al Qaeda is known as:

In formal communications Bin Laden has called the organization the International Front for Jihad against the Jews and Crusaders.