How to control 0 to 110 of AC with microntroller rms output of 0 to 5 v
First a word of warning: Please respect the 110V mains supply, it can be lethal.

Dimming of a lightbulb works like: Dimmer to bulb control

The below two links describe a method with a zero crossing circuit and a triac to control a lightbulb from Arduino (or most any microcontroller). The controller is synchronized to the mains supply by a zero crossing detector and in turn fires the triac at sooner or later, resulting in dimming the lightbulb.

Here is a project that explains a project to make just that with an Arduino.
http://www.instructables.com/id/Arduino-controlled-light-dimmer-The-circuit/step2/Arduino-controlled-light-dimmer-The-software/

Luckily this hardware is also readily built available, here is an example: http://www.inmojo.com/store/inmojo-market/item/digital-ac-dimmer-module/