Assuming the buyer paid by paypal, you can check their email adress there if you are the seller. I assume that you are not the seller though. If not, you can check to see if the buyer is also selling through ebay
No, what happened was there was an item I was interested in with a buy it now price and whilst I was thinking about it it got sold. I did email the seller and explain the situation and ask if he/she could let the buyer know I was interessted/or provide with his/her details but the seller didnt rely back. Typing in the buyer's code name doesnt work (as ballroom pointed out) , but the buyer must have a shop or be a business seller as he/she has the most number of stars I have ever seen, - which could help me find out the identity but I am stuck for the moment at what next to do.