Mycobacteria are very slow growing bacteria, so finding and identifying them isn't as easy as strep or staph germs.
In the AFB test – "Acid Fast Bacillus" or "Acid Fast Bacteria" – the sputum is spread on a slide, with a dye, and grown for a couple days to see if any bacteria appear there (Mycobacteria include TB as well as MAC.) If anything shows, a culture must be grown (4-8 weeks) to see what exactly the bacteria is, and then it is further tested (2-4 more weeks) to see which antibiotic(s) it is susceptible to.
If you have had a previous positive MAC culture, the lab may send the sample for culturing, even if the AFB test doesn't show positive, because treatment may have severely reduced the number of bacteria in your system.
At the same time, a portion of the sputum sample may be cultured to look for other bacteria like pseudomonas, as well as some fungal infections, since the symptoms may be similar.