Acid-fastness is a physical property of certain bacteria, specifically their resistance to decolorization by acids during staining

Staining (biology)

Staining is an auxiliary technique used in microscopy to enhance contrast in the microscopic image. Stains and dyes are frequently used in biology and medicine to highlight structures in biological tissues for viewing, often with the aid of different microscopes...

 procedures.

Acid-fast organisms are difficult to characterize using standard microbiological techniques (e.g. Gram stain - if you gram stained an acid-fast bacillus (AFB) the result would be an abnormal gram positive organism, which would indicate further testing), though they can be stained using concentrated dyes, particularly when the staining process is combined with heat.