An amino acid carries an amino group, a carboxyl group, and a side chain that distinguishes it from the others. Twenty are genetically encoded; synthetic peptides also incorporate non-standard ones, including D-isomers and modified residues that do not occur in translated protein.
Forming a peptide bond joins the carboxyl group of one amino acid to the amino group of the next and releases a molecule of water. What remains in the chain is therefore no longer the free amino acid, and is called a residue — which is why the arithmetic works out as the sum of the residues plus one water, not the sum of the free amino acids.
Side chains determine most of what distinguishes one peptide from another: charge, hydrophobicity, the capacity to form disulfide bonds, and the sites available for modification. Composition alone does not fix a molecule, though — the same residues in a different order are a different peptide with the same mass, which is why sequence is the definitive description.