If you are selling your home you may want to have it appraised before you put it on the market. It is money well invested. I have seen owners sell a home $20,000 to $30,000 under appraised value because of an Agent's Market analysis. As the seller you will never see the lender's required appraisal and if there is a major difference between the value and the selling price you will only know if it doesn't appraise high enough. Then of course you will have to lower your price or find someone willing to purchase with enough cash to make up the difference. (Fat chance) If it appraises a substantial amount higher, you will never know.
If there is any possibility that your home could be financed by a FHA loan then the appraiser you select should be FHA approved. Order a conventional appraisal, it's less expensive. But, if your buyer does get a FHA loan then your appraiser will be able to convert it. If the appraiser is not FHA approved then you will have to purchase a brand new appraisal. If your home does not qualify for FHA financing then this is not an issue.
Use an appraiser that is familiar with your area. Ask him if he is.
Your appraisal is only usable by a lender for 6 months. Some appraisers will update one they did for a small fee. If your appraisal IS usable by the buyer's lender the appraiser will usually charge a small fee to put it in the lenders name. Discuss these things with the appraiser you choose.
If you are a buyer it is absolutely best to let your lender order the appraisal. Some lenders have special requirements or a list of approved appraisers. You must stay within their guidelines if you want their loan. Sometimes if a seller already has an appraisal it can be used but sometimes it can not. If the seller is financing your purchase you may want to order your own appraisal. It just depends on your relationship with the seller.