なくちゃいけない

Meanings

usually written using kana alone

Used when something is essential and cannot be omitted. Often follows a noun marked with が or は.

なくてはいけない
無くちゃいけない is a casual, spoken contraction while なくてはいけない is the more standard, neutral form used in both speech and writing.
なければならない
無くちゃいけない sounds casual and conversational, whereas なければならない is more formal and often used for rules, regulations, or written obligations.
なきゃいけない
Both 無くちゃいけない and なきゃいけない are casual contractions expressing obligation, but 無くちゃいけない derives from なくては while なきゃいけない derives from なければ.

Contraction of なくてはいけない, from 無い (ない, 'not exist') + ては + いけない ('must not'). The ては contracts to ちゃ in casual speech.