A buyer's premium is effectively a service fee paid by the high bidder of an item sold at auction to the auctioneer. A buyer's fee of 10% would imply that if you are the high bidder on a lot that sold for $100.00, you would be responsible to pay $110.00 at check out. ($100.00 bid plus 10% for a total of $110.00).
Buyer's premiums were introduced by the larger NYC auction companies a few decades ago as a way to entice consignor's of high end goods to choose their company to represent them. The model proliferated out and is now practiced by the FAR majority of auctioneers. Buyer's premiums (in our area, outside Pittsburgh, PA) generally range from around 10% up to about 20%. Some of the largest auction houses now rarely charge a commission to sellers and can charge the buyer up to 40% in buyer's premiums!
It is very important to read the auction announcements slowly, not all auctioneers do a great job of making their policies clear, you want to be aware of your end responsibilities. It is extremely common for their to be just a one line announcement "15% buyer's premium". That can end up being a lot of money.
Many local auctioneers discount a couple of percentage points off of the buyer's premium for those paying with cash or check, as credit card service fees are not incurred.