Gomen nasai means "I am sorry" or "Sorry." This phrase is used to apologize for mistakes or to express regret. It's generally more direct and personal than "Sumimasen" when apologizing. You'd use it if you've done something wrong or inconvenienced someone. "遅れてごめんなさい！" (Sorry for being late!).

自(じ)分(ぶん)の間(ま)違(ちが)いや迷(めい)惑(わく)をかけたことに対(たい)して謝(あやま)る時(とき)に使(つか)います。親(した)しい間(あいだ)でも使(つか)います。「遅(おく)れてごめんなさい。」