Correct spelling


Correct spelling, explanation: the word apologies is the plural form of word apology (so we add suffix -ies and get rid of last y to create plural form). Apology originates in Green apologia. Therefore spelling with one p is the correct form: apologies. Appologies is a misspelling and there is no reason to spell this word with double p.

Definition of apologies:
nous, plural of apology, the act of showing regret
Apologies are unnecessary; you didn’t do anything wrong.
My apologies for being so late. Shall we start now?

Incorrect spelling


Incorrect spelling, explanation: appologies is an incorrect form, because there is no double p in the root word – neither in singular apology nor in Greek origin apologia. Therefore the correct form is apologies (plural of apology). Apologies is correct because single p is there just as in its origins.