Japanese individuals are well-known for not revealing or hiding his/her true emotion and intension. So if you happen to offend them and apologise, they might not truly forgive you even if they say "No problem". However, if you can say "I'm sorry" in Japanese, they will be touched to pardon you.
Casual
ごめん
Gomen.
Basic
ごめんなさい
Gomen nasai.
Basic and Formal
すみません
Sumimasen.
Most polite
申し訳ございませんでした
Mooshiwake gozaimasen deshita.
Osaka dialect
堪忍やでぇ
Kan nin yadee.
Hiroshima dialect
勘弁してつかあさい
Kanben shite tsukaasai.
Used by young
まじでさーせん
Majide saasen.
Followings are words that are used in not very serious situations and used to provocate opponent.
ごめんちゃい
Gomen chai.
めんごめんご
Mengo mengo.
わりーわりー
Warii warii.
