Aggregate base is a layer of compacted crushed stone between the subgrade and the pavement, under asphalt or concrete, that spreads the load and lets water drain out from under the surface.
It's usually a dense-graded crushed stone, often called crusher run or known by your state DOT's base spec number, that locks together when it's compacted. You buy it by the ton, spread it loose and compact it in lifts.
How deep it goes depends on the soil and the traffic, and the plan or local spec governs when there is one. As a rule of thumb, figure 4 inches under a driveway or sidewalk and 6 to 12 inches under a commercial lot or anywhere trucks turn.
To order it, work out the compacted volume and convert to tons. Compacted crushed stone runs around 2 tons per cubic yard, but ask the quarry. Six inches of base under 10,000 square feet is 5,000 cubic feet, or about 185 cubic yards, which is roughly 370 tons before waste.
Soft base is the classic paving change order. Milling or excavation exposes wet or pumping material that has to be dug out and replaced with stone before anything goes on top. That's why a good proposal carries a unit price for base repair, per square foot or per ton of stone, before anyone finds it.