1) to rejoice, be glad
2) to rejoice exceedingly
3) to be well, thrive
4) in salutations, hail!
5) at the beginning of letters: to give one greeting, salute
Part of Speech: verb
Citing in TDNT: 9:359, 1298
A primary verb; to be full of "cheer", that is, calmly happy or well off; impersonal especially as a salutation (on meeting or parting), be well: - farewell, be glad, God speed, greeting, hail, joy (-fully), rejoice.
These two sites give similar information, with the definition from several dictionaries and statistics on the use of the word.
Perseus Digital Library
University of Chicago's Logion lexicon
Thayer's dictionary plus other information.
From this site's dictionary (in Italian)
Click on the first column to search for that word as a form of the root χαίρω; click on the second column to search for that grammatical form of the root χαίρω; click on the third column to search for that word and grammatical form; click on the total to search for the root χαίρω.