First off, DON'T PANIC. An appraisal is an opinion of market value given by a licensed appraiser on a particular property at a specific point in time. The key word here is opinion. Appraising is not an exact science otherwise there would be no need for appraisers and all anyone need do is get a value from online services like Zillow.
Even though licensed appraisers all have to adhere to the same set of guidelines approved by the Appraisal Foundation, there are so many variables involved that the end result can only be described as an opinion of value. This does NOT mean that the appraiser's opinion is not a correct valuation; it just means that there can be more than one opinion that is correct. These differing values are supported by the appraiser's choice of considerations, and the weight given to each of them, which affect the house being appraised. The differences in appraised value can be even more marked if one of the appraisers is from outside the area where the property is located as opposed to the value given by a local appraiser.
What to do. If you had an appraisal done when you priced your home for sale, get in touch with that original appraiser and ask them to do an updated appraisal for you. Tell them that the value is coming in lower than the amount they appraised it for and you need documentation to support your price. You will probably have to pay for this, but the cost should be lower if the original appraisal was completed not too long ago. If the re-appraisal supports your sale price, get in touch with the buyer's lender immediately and present your case for this value to be accepted. If the difference in value is considerable, the lender might insist on a third appraisal and/or an appraisal review.
What if you didn't get an appraisal before you priced your house? Well then, you have a couple of options. Obviously, the first is to get your own appraisal done and hope that it comes back at the price you need and, if it does, then proceed as above. If this second appraisal also comes in lower, then it would appear you have overpriced your house to begin with. You can then either lower your price to the appraised value or ask that the buyer to come up with the difference in cash if they still want the house, or you could carry a second for the difference. Most buyers won't do this unless there was a burning desire for them to have that particular home. If you then decide that you still want to sell, you can put the house back on the market at the appraised price.
The other option is to challenge the appraiser's findings. This can be very difficult and time consuming. To do this, you will need a copy of the appraisal, not just the summary. The best thing to challenge is the comparable sales the appraiser used. Remember, these are homes that have sold and closed, not those still on the market. Look at these very carefully and then go visit them to see if they are truly comparable to your house. You need to make sure that the appraiser is comparing apples to apples. If there are comps used that are not similar to yours, maybe in a different tract, different and lower quality builder, a builders value priced model, different school district, in the county instead of the city etc. then you have a strong case for an appraisal review. Rather than just going to the lender and saying that the wrong comps were used, you have to go armed with comps that will support your claim of a higher value. To do this, you will need to go to the County Recorder's office and search for homes that are similar to yours which have recorded as sold within the past six months. The more recent solds are the best and most accurate ones to use. When you look at the appraisal report, look carefully to see what adjustments the appraiser made to the comps to compensate for amenities that you do or don't have in your house. Sometimes you might find incorrect allowances made for a/c, pools, spas, new roof, landscaping etc. These all have value but not the same value as the price you paid for them.
After reviewing everything carefully and you can't find any glaring discrepancies in the report, then the best thing to do is accept the findings and move on. If you sold your house using a real estate agent, then there are things that they can help you with, but that's another topic.