"Give me a spoon!"
Мне is the dativ (and also locative, btw) form of the personal pronoun я. The similar in English is the conjugation of I into me when the pronoun plays the role of the indirect object in the sentence (i.e. to whom is something happening/who is receiving the action in the sentence?)
Дай мне ваду! Give me water!
Since you could say "to me" in English (rather than only "me"), мне is sometimes translated that way. There is no notion of the verb "give" in мне, it could be any other action such as e.g. "show me", "teach me", ...
Subject (nominative) actions (verb) object (accusative) to indirect object (dative).
He gives water to me (or he gives me a present), He = [acting] subject (nominative) gives = action (verb) water = [direct] object (accusative) me = indirect object (dative)
Он даёт ваду мне (to keep the word order the same...you would normally turn it around like in English.)