I used to sell vinyl records as one of my main stock lines when I first started selling on ebay. . I  packed them in a plastic bag and then sandwiched them between two heavy, double core cardboard squares,( fridge boxes from electrical shop or car radiator boxes from auto mechanic ) cut slightly larger than the record. Tape around the outside. NOW THE IMPORTANT BIT ......It is vital that the core flutes run in opposing directions. This is what gives the package its strength. If you just fold a cardboard rectangle in half the flutes run the same way and the cardboard is very easy to bend, damaging the record. 

I have sent many hundreds of vinyl records this way and never had a problem. Using this method, they also usually go as an under 500 gm. parcel, saving on shipping costs. Some other methods suggested may push the weight over 500 gm. increasing postage costs substantially, making many ebay record sales unviable..  NOTE - not suitable for vintage 78 gramaphone records though. They are too prone to cracking for this method.