→ 〜なければいけない → 〜なければいけません
なければいけない means "must", "have to". Take the ない-form of a verb and replace ない with なければ, then add いけない: いかない行かない → いかなければ行かなければいけない. いけません makes it polite.
Literally it says "if I don't, it won't do": なければ is "if not", and いけない is "no good". The ない drops out entirely; only its stem stays.
The two overlap almost completely. なければならない sounds stiffer and is the usual choice for rules, duties and writing. なければいけない is more at home in speech, about what a particular situation calls for.
In conversation なければ shortens to なきゃ, and the いけない is often dropped. ないと works the same way.