Wrongness with だめ, いけない, ならない, and how to say “must”

だめ is a word that means something like "no good", "useless", or "no use". It can be used to say that a statement would be bad, after connecting that previous statement with ては or conditionals like たら, etc.

あまり迷惑をかけてはダメですよ。  
Inconveniencing others too much is not good. (lit: “if you inconvenience others too much, it is not good)

日本だったらダメなんだろうけど、ここでは大丈夫みたい。  
Had it been Japan it would’ve been bad, but here it seems fine.

ならない and いけない are similar to だめ, but they feel stronger and have an implication that something is prohibited, either by someone’s decision or social expectations.

他人のものを、盗んではならない。
Stealing other people’s stuff is unacceptable. (lit: “it is unbecoming”)

All three of these “not good” groups (だめ, ならない, いけない) are often used with a negative conditional statement (〜なければならない, 〜なくてはいけない, 〜ないとダメ, and any other similar combination) to convey a meaning of “must do”. Since Aてはならない means “must not do A”, then logically (not A)てはならない means “must not not do A” hence “must do A”.

This is how you say “must do X” or “have to do X” in Japanese. It sounds like a mouthful, but you’ll get used to it in no time.

やらなければならないことはたくさんある。
There are a lot of things I must do. (lit: “things that if I don’t do it won’t be acceptable”)
Style-wise, the combination of 〜ないと + ならない is seen as poor form and some people will consider it incorrect. You will come across it relatively often even in books, but it's advisable to avoid using it and instead opt for 〜ないといけない.

なければ is commonly contracted to なきゃ and ては is commonly contracted to ちゃ (では is じゃ).

死んじゃダメ！
You must not die! (lit: “dying is not good”)

When implied by context, at the end of a statement, the part after the conditional is regularly dropped because these expressions are so incredibly common that everyone understands what comes after anyway.

冷めないうちに食べないと。
You must eat before it gets cold.