Ask the seller why they are selling when you tour the house. If they say that they are just “testing the waters”, move on -you won't get a good deal. If, on the other hand, they explain that the house is too much for them or that their job is moving to another state, this can be an excellent opportunity for you to get a good price.
Appraisals can be misleading. They're only good for about a month. Look at the sale dates of the comparable homes listed in the appraisal (“comps”) -they should be 90 days old or less to be accurate. Don't trust “time adjusted market data”. Remember that appraisals are just someone else's opinion of what the house is worth. Also, appraisers don't thuroughly inspect a house, so they assume no defects unless they are obvious. You are entitled to a copy of the appraisal your lender uses. Look it over for mistakes.
Pay attention to the Replacement Cost estimate in your appraisal. This can forecast future trends in the housing market in your area. If the replacement cost is less than what the house is being sold for, new homes will be built in the area and will drive down the price resulting in a loss. If the price is 25% to 40% higher than construction costs, home values will stop increasing and it is a sign of a bad short-term investment. When a site's value exceeds 60% to 70% of the total value of the property, be careful, the house has lost its value. Buy the site for its lot value, but know that you won't get much back for any improvements you make to the house.
Make sure the “comp” houses that your appraisal is based on are really comparable to the home you are looking at. Check to be sure the school district is the same. There are often premiums paid on property that falls under a premium school district.
Remember when looking at prices of homes that have sold recently, other things could have been added to the loan (ie escrowed money for roof repairs, or the sellers could have paid the closing costs, etc) Look for homes sold by real estate agents -they usually better reflect market price (you won't be looking at a home that was sold by a father to his son for a fraction of the actual value).
Check how the city is doing financially. If they are operating at a large deficit and have been for awhile, property taxes are likely to increase and the services offered by the city are likely to decrease.