Learn Polish: Czy wybaczy mi pan mój błąd? - Will you forgive my mistake, sir?

The verb wybaczyć (to forgive) requires the Dative case for the person receiving the action. You are literally asking "Will you forgive to me my mistake?". Therefore, we use mi, which is the short Dative form of ja (I).

Why is the verb wybaczy in the "he/she/it" form when the English translation says "Will you..."?

In Polish, formal questions using pan (sir) or pani (madam) always take the third-person singular form of the verb. It translates to "Will you..." but grammatically functions exactly like "Will the gentleman forgive...".

Why is mój błąd (my mistake) exactly the same as its dictionary form? Shouldn't the object of a sentence change its ending?

It is actually in the Accusative case because it is the direct object (the thing being forgiven). However, błąd is a masculine inanimate noun. For these types of nouns, the Accusative form looks exactly the same as the Nominative (dictionary) form.

Could I put mi at the end of the sentence, like Czy wybaczy pan mój błąd mi?

It is highly unnatural to put short pronouns like mi at the very end of a sentence. Because mi does not carry emphasis, it prefers to "tuck in" right after the verb or the first stressed word. This is why Czy wybaczy mi pan... sounds much more natural.