When turned on, the monitor went blank (or black). Have re0boted PC and no change. We replace the LCD monitor with another one, and it works. What could be wrong with Acer AL1916W LCD Monitor that we bought 6-23-2006.

That is old enough to have bad caps by now, that will be the first thing to look for.
If you are going to DIY and already try the monitor with another PC, and have proper tools and know safety precaution then please read on:
Most common failures in the LCD monitors are bad capacitors (bulging top/seal or leaking) in the power supply (they should be replaced in a set), blown fuses; poor solder joints, failed inverter circuits (blown fuse, shorted transistors, shorted/open transformers), bad lamps (poor solder connections or worn out lamps). You will need to open it up and inspect the inside, see example of failed ACER monitors to get some ideas what to look for: http://s807.photobucket.com/albums/yy352/budm/
Post back what you see inside so we can guide you further and it will help out other people in the future also.

Basic LCD monitor and TV troubleshooting guide:
http://www.fixya.com/support/r6150077-basic_lcd_monitors_troubleshooting
http://www.fixya.com/support/r5093881-lcd_flat_panel_tv_troubleshooting_guide

Learn about bad caps: http://en.wikipedia.org/wiki/Capacitor_plague

Capacitors kit: http://lcdalternatives.auctivacommerce.com/ he can make you a set of caps for you.
Or www.digikey.com just make sure to use caps with low ESR, 105c, high ripple current, long life rating such as PANASONIC FM or FC series.