In Anthony Burgess's novella, A Clockwork Orange, it's the "nadsat" slang version of saying you're sorry. Synonymous with "apologies" as in "my apologies".
Oh, did I step on your foot? Appy polly loggies, O my Brother!
by vivovixivictum May 1, 2006
Nadsat for an apology.

Nadsat is the language used by the main character and his friends in clockwork orange.
by Marc November 30, 2004