"He does not want the hat."
Translation:Níl an hata uaidh.
Ní is used to make a verb negative, e.g. ní rithim (I don't run). Níl is like the negative of tá, e.g. níl me ag rith (I am not running). I suppose it's the equivalent of the difference in English where for verbs, you usually use do not for negatives, but with "to be" (I think it's the copula, but I'm not entirely sure what that actually is), you just use not (i.e. I don't run vs. I am not running). I hope that makes sense ^-^;
There is no trap. níl is the negative form of tá. Níl is never used in any other context, and it is never used with any other verb, because it is already a verb. All other verbs are negated with ní, and ní causes lenition.