Griffith's experiment was conducted in 1928 by Frederick Griffith which was one of the first experiments suggesting that bacteria are capable of transferring genetic information, otherwise known as the “transforming principle”, which was later discovered to be DNA.
Griffith used two strains of Pneumococcus (which infects mice), a S (smooth) and a R (rough) strain. The S strain covers itself with a polysaccharide capsule that protects it from the host's immune system, resulting in the death of the host, while the R strain doesn't have that protective capsule and is defeated by the host's immune system.
In his experiment, bacteria from the S strain were killed by heat, and their remains were added to R strain bacteria. It turned out that the formerly harmless R strain now was able to kill its host. It had been transformed into the lethal S strain, obviously by a transforming principle that was somehow part of the dead S strain bacteria.
Today, we know that the DNA of the S strain bacteria had survived the heating process, and was taken up by the R strain bacteria. The S strain DNA contains the genes that form the protective polysaccharide capsule. Equipped with this gene, the former R strain bacteria were now protected from the host's immune system and could kill it.
More detailed description
Griffith injected mice with two different strains of each bacteria (IIS, IIIS, IIR and IIIR) and observed that the mice injected with the strains with out the polysaccharide coat (mutant R strain) lived, while the mice injected with the polysaccharide coat (S strain) died.
IIR strain can mutate into IIS, and IIIR into IIIS and vice versa, but not IIR into IIIS and vice versa.
However, if the mice were injected with heat-killed IIIS strain pneumococcus, they survived. But if they were injected with both heat killed IIS strain and living IIIR strain pneumococcus, they died. Both IIS and IIS strains pneumococcus were isolated from the mice’s blood, suggesting that they could not have resulted from the mutation of IIIR strain pneumococcus.
See also: Genetics, Hershey-Chase experiment, Oswald Theodore Avery