The balloon sticks to the wall because of a process called induction. When you bring the balloon close to the wall, the balloon has a charge on it that is making the negative charges (which are tiny particles called electrons) in the wall move away from it. This happens even though the balloon doesn't touch the wall.
Because the electrons in the wall move away, the area of the wall closest to the balloon becomes positively charged (since all the negative charges moved away). The balloon, which is negatively charged, is attracted to this positively charged area on the wall.
This attraction between the negatively charged balloon and the positively charged spot on the wall is what causes the balloon to stick. So, even though they never touch, the charges create a pull that holds the balloon to the wall.