How do I evaluate the quality of a road bike? I am specifically looking at a Fuji Finest Womans road bike - 51cm.  I am not sure what year it is. The ad description says: 


  Women's specific geometry, 51 cm
  Shimano Sora, Ritchey stem, Aheadset,
  pedals with toe cages.  Great shape,
  smooth and easy-riding bike.


The asking price is $350 (Canadian).

I am curious about the quality of Fuji road bikes. How do I evaluate the quality of this road bike -- or any road bike for that matter?
To expand Stephen's answer a little, because there are a couple of really important checks to make. What you're looking for are signs that the bike is being sold because it's not worth fixing:


check the frame and fork for damage. Specifically, if the frame has crash damage it is worthless. Look for slight bends in tubes that should be straight, and for places where the paint has flaked off because a tube is bent. Look at the angle between the front fork and the head tube of the bike, and the gap between the front wheel and down tube. Bikes that have been crashed into something generally bend either the fork or the down tube just behind the head tube. From the picture that bike looks fine.
Look at the braking surfaces on the rims. If there is a wear line on the rim check that, otherwise run your hand along the rim and feel for a curve (across the short side from the brake pads wearing into it), or a break in the rim. New wheels are expensive, so if the rims are too worn it might not be worth buying the bike.
Look at the rest of the bike as Stephen suggests - does everything look ok, do the brakes feel smooth when you test them, is the chain clean, are the tyres in ok condition.
now ride the bike. Does it shift smoothly but definitely? Does it run quietly in every gear? Does it shift under power? Do the brakes work?


Most of the last points are things you can usually fix easily and cheaply. Which means that if the seller has not done them, it could be because the problems are not cheap. Rather than new gear cables and outers, you might be looking at new shift levers (which are not cheap). I would not bother looking at chain wear because even a new chain and cassette are cheap enough that it won't matter, and expecting the seller to throw out a half-worn set just so you can have new ones is wasteful.

Also, if you're not sure it is worth while taking it to a bike shop for a check-over and quote for repairs. Offer to pay them a token amount because you're not getting the work done you just want to know what they think. If the seller doesn't want you to do this before you buy, do it immediately after. That way you can (probably) still go back for a refund.

Please don't be scared off by the list of things that might be horribly wrong. Most of us have bought second hand bikes many times and ridden them a lot. It's that 1% chance that you're getting a lemon that we're talking about, not some mythical horde of dodgy dealers.