。
Don't forget to mail my letter, okay?
Literal
Not-forget-[without-ず] my [possessive-の] letter [object-を] send-[request-て] [tag-ね]
出す literally 'put out' is the everyday verb for mailing a letter — alongside the more formal 投函する ('drop in [the postbox]'). 私の手紙 pins the letter as belonging to the speaker — a small detail but a good example of how possessives fill in where Japanese could otherwise leave the argument vague. The te-form + ね softens this from a command into a friendly reminder, the kind of thing a parent or partner would say on the way out the door.