Overly simplified, an AC dimmer switch divides the incoming power flow into slices. As you turn the dial 'up' the slices that are allowed to pass through to the lamp become wider. At full brightness the adjacent (in time) power slices are close to 100%

Turning the dimmer dial towards more dim delivers narrower time slices of power flow with proportionally larger time segments of 'no power flow'. The unneeded power is not wasted, it simply does not flow at all.

This is electronically akin to very rapidly (120 times per second in USA and Mexico) flipping a light switch on and off. Adjusting the ratio of on time to off time changes the amount of average power arriving at the lamp.

There is some modest inefficiency in the electronics doing this rapid power switching, which is why a 'dimmer switch' gets mildly warm. Better quality electronics within the dimmer can reduce the amount of waste heat.

Getting more technical, the dimmer cuts slices away from each half cycle of the incoming alternating voltage sine wave. For an inexpensive dimmer the resulting waveform delivered to the lamp is a hacked up voltage, no longer 'clean'. This is why some bulbs will buzz or hum when operated on a dimmer, but will be quiet when operated using a regular on-off switch.

Further complicating matters, bulbs that are not incandescent, such as CFL, low voltage halogen, and now LED bulbs, often are not compatible with a simple Triac dimmer. These non-traditional bulbs may not dim smoothly, may flicker, buzz or even overheat internally if used with an incompatible dimmer.

There are special Electronic Low Voltage, Magnetic Transformer Low Voltage, and other dimmer switch designs which produce more controlled waveforms. These typically cost more, and often enough are not widely stocked on retail shelves.