The word ごめん (gomen) essentially means "forgive" or "pardon". ごめんなさい (gomen nasai) is a common phrase that literally means "forgive me" or "pardon me", and ごめん (gomen) or ごめんね (gomen ne) is a more colloquial way to say the same thing. Like saying "sorry" instead of saying "I am sorry".
I don't know much about samurai-era Japanese, and so I've never heard "kiri-sute gomen" before. But assuming it's written 切り捨てごめん,
then it would presumably mean something like "forgive them by cutting them and throwing them away".
Based on the defintion Darius Jahandarie linked in the comments, The meaning of "Kirisute gomen" is more like "pardoned for cutting and throwing away" and refers to the special right of being able to kill someone of a low station and not be charged with a crime. Essentially, a "license to kill" (legal pardon for killing). According to the same article, it was not a general license to kill in every situation, but was to be exercised in accord with the warrior customs of the time.