Why do my lights dim, when my neighbor's central AC turns on?
You and your neighbor may very well be on the same power mains feed. Any heavy current draw on one branch of the circuit will be seen as a voltage droop back up the line toward its source. As that droop passes the point where your mains connection joins in with the neighbor's mains your branch will see a corresponding droop in voltage. This can lead to a dimming of your lights. 

This type of voltage droop will always exist in power feed circuits due to the resistance in the wiring. The designer of a power distribution system simply selects wiring sizes with low enough resistance (i.e. fatter sized wire) to keep the droop small enough that it does not create annoying and functional problems with devices attached to the power source. It would appear that in your case the power feed system is not designed with large enough wiring to keep droop to a tolerable level for the loads it is expected to supply. Often the case is that a power distribution network may be designed for a certain size of expected loads. Then later someone comes along and adds heavy duty loads that go way beyond the designed to capacity of the circuits.