Can you please help me with finding differences between ごめん and 許す in "forgive me" meaning? Does "許す" means something more informal than "ごめん" and can be used with someone close (friend, family)? Or maybe "許す" have more regretful meaning than "ごめん" (I often see this word in manga in "sad" situations (like in someone's death))? As "I'm sorry" i often see "許す" in "許せ" form. I found, that "せ" makes imperative form of the verb and also i saw somewhere that it's a rude form, while "ごめん" is a polite. So please help me to figure this out) Lots of thanks to everybody!
許す means to forgive (or allow/permit/tolerate). ごめん means "sorry."
許す by itself doesn't mean "forgive me."
Maybe you meant 許してください?
If so, 許してください = "forgive me" and ごめん = "sorry."
(As with all things language-related, there are about a million other things to be said on this topic, but judging from your level maybe this is the clearest explanation for now. For example, 許せ would probably mean person A is telling person B, who is in some way A's inferior, to forgive a third person C, although other situations are possible.)