Goodbye

Good-bye is a familiar interjection on parting in English. (The noun 'a goodbye' is best unhyphenated, but the interjection is best with the hyphen.) Although it may be found written without a finel '-e', AWE recommends that it should always have it, either as noun or interjection. In American English, on the other hand, good-by (interjection) and goodby (noun) are accepted variants.

Bye bye, or Bye (preferably to pedants 'Bye), are common contractions. Although they are useful in recording or imitating spoken dialogue accurately, they should be avoided in academic writing.

Etymological note: good-bye is a contraction (written in this form since the eighteenth century) of a fifteenth century formulaic utterance "[may] God be with you". This exists in many different forms in the works of Shakespeare, his contemporaries and writers generally until the nineteenth century: "godbwyes", "God be wy you", "god b'(o)y you", "god buy (or buy') you", which has suggested to some scholars (OED says erroneously) that the phrase meant 'God redeem you'. There is considerable variation between 'you' and 'ye'.

You may want to see Buy - by - bye, and maybe AWE's page on the equivalent word of greeting hello.