[subject] + 给 + [person] + [thing] [subject] + 给 + [person] + [verb] + [object]
给 as a verb means "give". It takes two objects in a row: the person, then the thing.
The recipient comes straight after 给. English can say "give a book to you"; with 给 the thing never goes first.
不给 is a refusal; 没给 says the giving didn't happen.
In front of a verb, 给 marks who the action is done for or to. 给 + person goes before the verb, not at the end of the sentence as "for him" does in English.
Some verbs take 给 straight after them: 送给, 卖给, 还给, 借给. The recipient follows, and the thing usually moves in front with 把 or goes last.
给我 on its own is an instruction to hand something over, and sounds abrupt without 请.