It's a contactless, disposable RFID card (that means it has an electronic circuit in it) good for five fares on almost any means of public transport in Istanbul—if the turnstile has been adapted to read the RFID chip, which few have (as of April 2010).
The beşiBiryerde card replaces the traditional paper single bus tickets, and supplements the Akbil electronic transit pass.
The beşiBiryerde's incredibly clunky name (BESH-ee-beer-yehr-DEH) means "five-in-one" in Turkish: five usages (beş kullanımlık) on one non-refillable card. (For a refillable card, get the Istanbulkart.)
The chip in the beşiBiryerde is supposed to be able to communicate with the turnstile up to a distance of 10 cm (4 inches). The company that developed the card would like to think this allows you to walk through the turnstile with the card in your pocket or purse and not have to pause or stop at the turnstile to register your fare.
Of course, this only works if the turnstile has been adapted to read the card, which few in Istanbul have so far.
In practice, I think few people are exactly the right height for their pockets to pass so close to the card reader. I think you'll probably have to put the card close to the turnstile, either by having it in your purse or wallet and putting that close to the turnstile, or having the card itself out and available and touching it to, or waving it near the turnstile RFID point.