Amino acid, as a term, means two things, with one use being a very small subset of the first. Chemically, as described above, an amino acid is a substance that has an amino group and a carboxyl group. Technically speaking, the term could probably be used in an even broader sense, I imagine that a molecule with a sulfuric acid group could also be called an "amino acid", but that would probably confuse people. But even the narrower definition of an amino group and a carboxyl group is much more broad than the usual usage of the term "amino acid", which refers to the 20 amino acids that are used to make up proteins. These 20 are distinguished by their "R-Group", an group of carbon chains or rings that attaches to the backbone of the amino acid.
There are theoretically an infinite amount of amino acids, since molecules could be made arbitrarily larger and still have those definitive functional groups on them. But even if we confine ourselves to amino acids the sizes of the ones that make up proteins, there are many different varieties of amino acids that could be used. There are many different organic molecules, and many different substitutions that can be made for them. For example, Phenylalanine has a Benzene Ring as its R-Group, but there is no intrinsic reason it couldn't include a Pyridine Ring instead. Or, for that matter, two linked Pyridine Rings. This is only one possible change, but by including other changes within it, for example adding methyl or hydroxy groups to various position on the different ring substitutes, the different combinations quickly become factorial. It is not that hard of a concept to understand, since it is very analogous to lego blocks: even with only a small amount of pieces that connect in seemingly small amount of ways, you can quickly reach millions of possible combinations.
So out of the millions of possible amino acids that could be used, why does life on earth only (normally) use 20? If, as Steven Jay Gould said in Wonderful Life, we could "rewind the tape" and let evolution proceed with a few different random occurrences, would we be living in a world where there was 32 amino acids, and three of them had Phosphorous atoms? Very possibly so. Which isn't to say that the current crop of amino acids is either inefficient or totally random, just that it is only one of many possible combinations that could be used. The basic reason that amino acids have different R-Groups is that the shape and electric charge on each one is different, which means that the amino acid, when incorporated into a protein, binds differently to different molecules, based on its shape and electric charge. The electric charge has to do with the varying electron affinities of Oxygen, Sulfur and Nitrogen atoms. In other words, even though amino acids could incorporate Bromine and Phosphorous, they don't need to. The twenty amino acids that we have work just fine for a great many functions.
To use an analogy, if were to rewind to the beginning of human written language, and reevolve the alphabet, we would probably still end up with some letters the same, such as the I and the O. Just like any set of amino acids would probably use alanine and glycine. However, we could very well have a Q that pointed the other way, with a dot in the middle, just like we could have a Tyrosine with an attached methoxy group. The main thing would be to have a set of symbols that were differentiable and combinable enough to make more complicated patterns.
This is of course, speculation, and a question that might not be answered until we find out if there is other carbon based life out in the galaxy. To return to a more prosaic level, the only thing that has to be remembered that "amino acid" has a technical meaning in chemistry, and a technical meaning in biology, and that people usually use it to refer to its more limited, biological meaning.