Learn Hardcore Russian: Пожалуйста, дайте мне ещё одну салфетку.

Дайте is the imperative form of дать, which means to give. The imperative is used for requests and commands, so here it means give (please).

This form is very natural in Russian when you are asking someone to hand you something.

Why does дайте end in -те?

The ending -те makes the imperative either:

plural: when speaking to more than one person, or
formal/polite singular: when speaking respectfully to one person

So in a situation like asking a waiter, cashier, or stranger for a napkin, дайте is polite and appropriate.

The informal singular version would be дай.

Why is мне used instead of я or меня?

Because Russian uses the dative case for the person who receives something.

With дать (to give), the pattern is:

дать кому? что?
give to whom? what?

So:

мне = to me (dative)
салфетку = napkin (direct object)

That is why мне is correct here.