Actually, an appraiser's responsibility as defined by Uniform Standards of Professional Appraisal Practice (USPAP), is to determine the Current value of a given property based on several factors, not simply past sales. Any sales data used to help determine value needs to be recent, preferred is less than 90 days old, but can be longer if limited sales in the area have occurred. The appraiser also has a responsibility to understand the current market trends on the effective date of the appraisal, and to assess the availability of financing, job growth (or decline), and any other factor that will help in assessing the Current value (not historical value).
The appraiser even has a responsibility to determine the value based on a reasonable marketing time for the property, rather than a forced sale, or an extended sales cycle (unless that is typical).
While value is eventually determined by a willing buyer and a willing seller, that has it's own set of issues if either of the parties are more "informed" than the other. In nearly every case, the buyer is the one who is least informed. For example, feeling comfortable with your purchase can occur for a number of reasons, not the least of which is you not knowing something important about the properties true value.
An appraiser is looking at past sales, but the real market value is determined by a willing buyer and seller. The object for the buyer is get the property for the lowest possible price and the seller the highest. The most important things for you in this situation is that you feel comfortable with the price that is offered and it works with your financial situation. I hope that this helps.
As long as you are comfortable with the appraiser's results (that is, you understand how they came to their conclusion of value), then any offer at or below the appraised value is where you want to end up.
Normally, an appraiser is going to give you a range of value, since the process of appraising is not a science, there's no way to defend a particular dollar amount, but they are also required to pick one number to give you nonetheless.
I'd suggest you target to end up (after your negotiations are complete), at no more than the one number that the appraiser provides, but you would be better off in this market to be below that number, somewhere between the bottom end of the range they provide, and the one number that they will have to give you.
For example, if you have a home that the appraiser's single point number says is worth $500,000, they will likely give you a range of value, say $480,000 to $510,000. Therefore, you should try to shoot for something in between, and if it were me, I'd shoot for something very close to $480,000 as my final number. That means I need to start negotiations by offering something in the neighborhood of $460K to $470K.
Hope this helps,