How can I stop my lights dimming when my Air Conditioning turns on?
First I would check the electrical connections to the compressor and the breaker in the panel, see that they are tight and not corroded. Next I would measure the starting current of the compressor with a clamp on ammeter. I had a problem like this on a friends AC, only his problem was tripping the breaker at times. His compressor is about 10 years old, and is harder to start now. When I measured the starting current it measured 175 Amps! Which on this compressor is the locked rotor current! We replaced the starting capacitor and added a "Hard start kit". Now the starting current is a respectable 50 Amps.

Some tech info:
The induction type motors used in these compressors have an extra motor winding. This is called a "start" winding and is used with the "start capacitor" it is there only to get the motor to initially turn and a mechanical switch disengages it at a low speed. What happens over time is that the motor load is increased due to wear, and the start capacitance becomes weaker. So why not just build a compressor with a large start up capacitance to begin with? Some higher end units do just that, but most don't to save $$$. Contrary to popular belief, the start capacitor size does not "Give a bigger jolt" to the motor, but delays the 60 cycle feed to the second winding fooling the motor to think it is driven by a poly phase feed. The larger the capacitor value, the bigger the delay - hence the lower the starting current. Without a start winding, the motor will sit there humming until it trips the breaker. Three phase motors (poly phase) do not have start windings because they have three windings.