What are 'acid stable' amino acids?
The amino acids asparagine and glutamine have hydrolysable amide groups on their R groups, as shown here:





Note the leftmost amide group on both amino acids. When exposed to acid, these groups would hydrolyse, releasing ammonia. 

This was of interest when people used to determine amino acid compositions by acid hydrolysing purified proteins (example paper which does this) and then running them through HPLC or mass spectroscopes. This method, however, fails to distinguish asparagine/aspartate and glutamine/glutamate due to their acid-unstable nature. 

So to answer your question, an acid-stable amino acid is any amino acid that does not degrade under acid treatment. 

Additional amino acids that degrade under such treatment can be found here, and includes serine, threonine, tyrosine, tryptophan and cysteine in addition to the two above amino acids. These amino acids degrade by alternative pathways distinct from the one by which the above amino acids degrade. 

Finally, assuming you are talking about amino acids in dietary supplements, the reason acid-stable amino acids are reported is because acid hydrolysis is usually the method used to hydrolyse the proteins into amino acids, and therefore the manufacturer would not be able to produce acid-unstable amino acids by this method, leading to that label on your supplements.