In order to cell cells and cellular detail under a light microscope, the sample must generally be stained first. "Acid-fast" organisms are organisms that resist decolourisation ('decolorization' in American) by the dilute acids solutions that are added as decolourants in many staining procedures. ('Fast' here doesn't mean 'quick', but is rather derived from the same root meaning as in 'fasten').
The chief example of an acid-fast bacterium is Mycobacterium tubercolosis - the bacterium that causes TB. Other examples include the organism that causes leprosy (Mycobacterium leprae).
The reason for being "acid-fast" is that these organisms have a high mycolic acid content in their cell walls. This same cell wall constituent also makes them very difficult to stain - you can't use the conventional Gram stain like normal. But once they are stained, they are very difficult to unstain.
While there aren't obviously direct pathogenic consequences of an inability to decolourise with weak acids, the mycolic acid is considered central to TB's ability to avoid being stamped out. The thick waxy cell wall that it helps to create cocoons the bacterium, protecting it from much of the arsenal that our own immune cells can muster. It also makes it difficult to eradicate with certain classes of antibiotics.