Azalea NaiSmith, an Asteroid Brain in the Tau Ceti system
The A-brain is, in short, a small body (smaller than a Moon Brain) that has been partially converted to computronium. The term is something of a colloquialism because unlike its larger brethren, the Moon Brain and Jupiter Brain, an Asteroid Brain rarely converts the entire body into computronium. This is because the mass is insufficient for an S3 intellect (at least with lower toposophic computronium) but usually far more than enough mass for an S1 or S2.
Instead, the unconverted asteroid (or comet, or moonlet) mass is typically reserved for industrial purposes. A classic A-brain, as opposed to an AI's asteroid facility, will thoroughly integrate the entire mass into its sensorium as if it is an extension of its body. In other terms, the A-brain is a mini-ISO that an AI will gradually develop to suits its needs and whims. Many seem satisfied in joining in computational complexes around gas giants or in asteroid belts, such as Kenehan-LeRiche complexes, while others develop into mobile vessels to explore the galaxy.