てくれる

This is a way of requesting someone to do something for the speaker or those associated with the speaker. The plain form is used for close relationships, with people of equal or lower status than the speaker.
「...てくれないか」is used by men. On the other hand, 「...てくれませんか」creates a more polite feeling than 「...てくれますか」. This expression is also used to draw the listener's attention.