なくてはいけない

なくてはいけない means "must", "have to". Take the ない-form, change the final い to くて, and add はいけない: 食べない becomes たべなくては食べなくてはいけない. Word for word it is "if you don't, it's no good". The polite form ends in いけません.

なくてはいけない or てはいけない

One negative more or less flips the meaning. てはいけない, on the ordinary て-form, means "must not".

なくちゃ

In casual speech なくては shrinks to なくちゃ, with or without いけない after it.

Register

なくてはいけない is neutral. なくてはならない and なければならない mean the same and sound more formal – they are what rules and written instructions use. In conversation, ないといけない and the short forms are more common.