It's not a verb, it's an short form of adjective. 'A good jacket is neccessary to me'.
Мне надо is only used with infinitives, you can't use it with a noun.
With infinitives, both are OK:
- Мне надо/нужно купить куртку.
But with nouns, you can only use нужен/нужна/нужно:
- Мне нужна хорошая куртка.
Yes, the plural form is «нужны».
Verbs always take нужно, but it's technically not a neuter form. Traditional grammar analyses надо and нужно as adverbs (and this is how we analysed it at school), while many modern scholars identify them as a separate part of speech called 'state-category words' (слова категории состояния) or 'predicatives'.
The difference between «надо» and «нужно» is negligible. «Надо» is a bit more colloquial.
It's the gender of the noun. Ку́ртка is feminine. Нужна́ is a short form of adjective, so it needs to agree with ку́ртка in number and gender. Grammatically this sentence is like "To me, a good jacket is neccessary."
Note that ну́жно can be either an adjective or an adjerb-like-thingy. When it's used with a noun, it acts like an adjective and changes its form.
When it's used with an infinitive verb, it's translated 'it is neccessary', and it doesn't change (мне ну́жно купи́ть ку́ртку 'I need to buy a jacket', literally 'to-me it-is-neccessary to-buy a-jacket'). Its exact function is a matter of dispute, some people consider «ну́жно» to be an adverb, others consider it a separate class of words called 'state-category words'.
«Я должна купить хорошую куртку» is grammatically correct, but должна (or должен, if you’re male, as должен/должна agrees with the subject) refers to obligation: «I must/should buy a good jacker».