I think you meant: 私は彼に謝らないといけません。
いけません is the negative polite verb form of いける, which means something "will do" or "is OK" or "will work".
"I think that will work." = いけるかもしれません。
いけない・いけません = "won't do", "you must not do", etc.
So, 「私は彼に謝らないといけません。」 literally means, "If I don't apologizes to him, that won't do". Of course, nothing makes sense translated literally, and it actually means, "I need to apologize to him."
A number of similar grammar structures exist in Japanese that consist of a verb in the negative form followed by the connector と, or the conditional negative verb form ~なければ, or the negative verb form nなくて followed by は, and finally a negative phrase.
???ないとだめ
???ないといけません
???なければ、なりません
???なくてはいけません
謝らないとだめ
謝らないといけません
謝らなければなりません
謝らなくてはいけません
I've always thought that it was interesting that a double negative becomes a "must do" in Japanese; saying "not doing X would be bad" is a common way of saying "I (or you) have to/should/must/need to do X".
謝る必要がある is the more literal translation from English (I need to apologize). This isn't grammatically incorrect, but I don't hear it used as often, and it has a slightly different nuance than the other phrases above.