ませんか (or ないか more casually) is an expression that is used after verbs as an invitation. It basically translates to ‘won’t you do (A) with me?’.
This expression is a little bit different to ましょうか, in that it sounds directly like in invitation to do (A). ましょうか tends to sound more like a suggestion that (A) should be done.

• 明日（あした）は温泉（おんせん）に行（い）きませんか。
  Won't you go to the onsen with me tomorrow? (Polite)
• 来週（らいしゅう）は釣（つ）りをしに行（い）かないか。
  Won't you go fishing with me next week? (Casual)

It is recommended to use the ませんか version, as ないか would be a bit too casual to use with someone that you are not very close with (and is also technically not correct Japanese).
To make it clearer that the speaker wants to do (A) ‘with’ someone, expressions like 一緒（いっしょ）に ‘together’ are often added, for clarification.

• 一緒（いっしょ）にお祭（まつ）りへ行（い）きませんか。
  Won't you go to the festival with me?

To examine the difference between ましょうか and ませんか, let’s have a look at how the nuance changes in the same type of sentence.

• ラーメンを食（た）べに行（い）きましょうか。
  Shall we go eat ramen?
• ラーメンを食べに行（い）きませんか。
  Won't you go eat ramen with me?

Here, ませんか will come across as an invitation due to the fact that you are asking the person if they ‘wouldn’t mind doing (A)’ with you. This is the main reason that it is considered a more polite option when inviting someone to do something, than ましょうか.