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1.
A quick and curt way to end a conversation, putting a thin polite spin on the rude abruptness.

Most uses of this phrase are referencing actor Gene Wilder's portrayal of Willy Wonka in the 1971 movie "Willy Wonka & the Chocolate Factory," where Wonka informs Charlie he has lost the contest because he drank the burp soda, a breach in the contract he signed. "You lose! Good day, SIR!" Wonka screams at Charlie before turning back to his desk.

Linking to an animated gif of the scene with the audio intact is a popular way to assert you are done with an argument on the internet.
"You lose! Good day SIR!"
by Billions June 19, 2007
 
2.
The gentleman's "Fuck you motherfucker"
Unclassy: Fuck you man, I'm out!

Classy: Well, I never! Good Day Sir!
by thisguyoverhere September 17, 2011
 
3.
One day, two marijuana enthusiasts smoked some rather potent weed. This weed did things to them like nothing they had ever smoked before. At the end of the night one of the stoners dropped off the second stoner, but before driving away he shouted "Good Day Sir" and drove off without saying another word. Ever since that day stoners occasionally use this expression in place of saying "goodbye"
DMS: See ya man, I'm out.

Jeff: GOOD DAY SIR!

DMS: Good day.
by Dank Master DMS February 13, 2005