What is the difference between transformation and transfection? How do both of these methods work?

Basically transformation came to be used to describe experiments in which the phenotype of an organism was changed by the uptake of DNA, and because of the way this developed in bacterial systems this DNA was usually a plasmid. Then it became possible to use purified phage DNA whereupon the 'transformed' cells produced phage particles - for obvious reasons this was called transfection.

When efforts turned to getting the same techniques working with cultured animal cells many of the vectors were based on viral genomes (e.g. SV40) and these workers referred to this as transfection. At this point the distinction between the two terms became meaningless, and usage tends to be for historical reasons. Interestingly in molecular manipulations of yeast, where there are no viruses, the transfection word is rarely used.