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,