Etymology is from the cartoon "Popeye", where the character Wimpy would frequently utter this phrase. He was a glutton, and would consume burgers at a ferocious rate but could rarely pay for his habit.
The phrase implies the underlying feeling that the person will unlikely actually pay for the hamburger (or whatever) on Tuesday (or ever, for that matter.
Bob: Aw man, I forgot my wallet! Hey Joe, I'll gladly pay you Tuesday for a movie ticket today...?
Joe: Yeah whatever...I'll buy your sorry ass another movie ticket just shut up.
An appeal for a small loan that will not be payed back.
Generally used by a person who considers themself a likeable egomaniac.
Random dude- "Gee thanks."
Server dude- "Who's paying?"
Wimpy- "I'm buying. He's paying."
Amber&Emily: She will gladly pay you tuesday for a hamburger today.
Innocent Bystander1: What in the hell does that mean?
Innocent Bystander2: Nobody really knows, they just sing it every day to piss off the lunch ladies.