This is by far one of the best bars in Istanbul, and one of the best I have ever visited anywhere. The name is Arka Bahçe and it is a hot hangout for all the students of Istanbul, especially those of the nearby Boðaziçi University, which I attended for a summer. Arka Bahçe (pronounced Arka Bachay) is tucked back in the narrow streets of RumeliHisarý and is easy to miss if you aren't paying attention. You can take a cab there, or take a cab to the South Campus of Boðaziçi and walk from there. From the main green of Boðaziçi follow the road down until you run into a gate, go through the gate and turn left. As soon as you turn you will see stairs on the right, go down the stairs and follow the road through another gate, take your first right, and you will be right there. The bar consists of an outdoor patio hidden among trees, as well as a small pool, and few fire escape type stairways leading to balconies above. There are tables, but most of the seating is found on the large pillows and beanbags strewn about the ground. These are great for relaxing with a drink, cuddling up with a date, or for a relaxing evening with a group. In the background chill alternative music such as Radiohead plays in the background. Arka Bahçe is the perfect place to kick back and waste away the night drinking and chatting with friends. A true treasure in the heart of Istanbul.
Also, a quick side note, the Sunday brunch is quite good as well.