In biochemistry and molecular biology, a residue refers to a specific monomer within the polymeric chain of a polysaccharide, protein or nucleic acid. For example, one might say, "The histidine residue is considered to be basic due to its imidazole ring." Note that a residue is different from a moiety, which, in the above example would be constituted by the imidazole ring or "the imidazole moiety".
Note the origin of this usage: during the process by which monomeric building blocks (e.g. amino acids) are strung together into a polymeric chain (e.g. a protein), some material (typically adding up to one molecule of water) is discarded from each building block, and only a "residue" of the building block ends up in the finished product.
For example, a residue is an individual amino acid in a peptide chain.