The term can be a bit ambiguous to many. For the sake of time and post-space, amino acids are what makes up proteins; when your body breaks down protein it becomes amino acids. There are many different types with many different purposes. For the most part, amino acid supplements will be branched-chain amino acids (BCAA) and less often essential amino acids (EAA).
BCAAs are made of 3 amino acids: valine, leucine, and isoleucine. With many reported benefits, they are used and recommended often. Some alleged benefits are: increased muscle protein synthesis (MPS), decreased muscle breakdown (catabolism), enhanced recovery, and improved immune function. While there are many conflicting studies, mostly due to poor study design, the consensus is that while beneficial in some aspects, they are not as great as manufacturers and retailers would have you believe.
MPS is not significantly affected by bcaas, nor is catabolism. The reasoning for this, in simple terms, is that absorption of amino acids into muscle is a natural process (with up to 70% of catabolized muscle being reabsorbed naturally) and can only be enhanced to a certain degree.
While pure athletic performance has yet to be observed in a quality clinical trial, recovery from exercise has been observed along with better immune function. Bcaas boost recovery by increasing peripheral blood mononuclear cell proliferation, which basically means that cell damage caused by intense exercise is mitigated with bcaas. Bcaas also change the way cytokines (signaling proteins) are produced post-exercise which reduces the stress placed on the immune system.
One caveat to the above: most studies on bcaas look at the combination of amino acids listed. There are a few studies showing that leucine alone, in higher doses than found in most bcaas, can be beneficial to performance and recovery. Another amino acid that is overlooked is tyrosine. While not as important for athletic performance, tyrosine is essential for production of L-DOPA, dopamine, epinephrine, and norepinephrine. Because of this tyrosine can help with depression, acute stress, and narcolepsy; general mood improvement can be expected with tyrosine.