They were the first set of nonsense mutations to be discovered, isolated by Richard H. Epstein and Charles Steinberg and named after their friend and graduate Caltech student Harris Bernstein, whose last name means "amber" in German (cf. Bernstein).
Viruses with amber mutations are characterized by their ability to infect only certain strains of bacteria, known as amber suppressors. These bacteria carry their own mutation that allows a recovery of function in the mutant viruses. For example, a mutation in the tRNA that recognizes the amber stop codon allows translation to "read through" the codon and produce a full-length protein, thereby recovering the normal form of the protein and "suppressing" the amber mutation.
Thus, amber mutants are an entire class of virus mutants that can grow in bacteria that contain amber suppressor mutations. Similar suppressors are known for ochre and opal stop codons as well.
tRNA molecules carrying unnatural aminoacids have been designed to recognize the amber stop codon in bacterial RNA. This technology allows for incorporation of orthogonal aminoacids (such as p-azidophenylalanine) at specific locations of the target protein.