Acid-fast and Gram stain are two different types of staining technique.
Bacteria that stain with acid-fast techniques are called "acid fast bacteria." Bacteria that don't stain with acid-fast techniques are called "not acid fast." At the same time, bacteria can be "Gram-positive" or "Gram-negative." For example, Nocardia is a bacterium that is acid-fast and Gram-positive.
Usually the reason people care about these designations is in order to figure out what species a bacteria is. The issue usually arises in the context of a patient's sample that has grown bacteria. At first the doctors won't be sure which bacteria it is. They'll have guesses based on how the patient presented, but they won't know, and they want to know because that will help them pick the best treatment.
A first step toward "speciating" the bacteria is to do a Gram stain on it. This is because for historical reasons determining whether a bacteria is Gram-positive or Gram-negative goes a long way toward speciating it. Acid-fast staining is less common, and is used mostly for diagnosing tuberculosis (caused by Mycobacteria, which are acid-fast bacteria).
There are details about the cell walls that determine whether an organism will stain with Gram stain (thick walls without mycolic acids) or acid-fast, but hopefully this answers your question.