In an East meets West kind of sense?
It's a cliche but yes, that's definitely a huge part of it. You can get that feel in parts of the Balkans, Lebanon and Israel too (and I'd assume Russia though I've never been), but not quite as natural (if that makes sense?) as in Istanbul. Then there is the magnificent setting on the Bosporus and Golden Horn, the sheer scale of the place, the weight of its history, the food, the vibrant buzz, the people. I could go on. It all adds up to a unique place.
Any places you specifically recommend? Been thinking of a planning a trip there for a while now.
All the obvious tourist sites around Sultanahmet and the Grand Bazaar are worth seeing, as are the more modern neighborhoods around the main shopping street Istiklal Cadessi and Taksim Square with the bars and restaurants. Most tourists on a short visit only really get to see these two areas, as apart from the sites they're where most of the hotels are. Apart from those:
Eyüp is a neighborhood at the top of the Golden Horn centred around a mosque/shrine dedicated to a companion of the Prophet Muhammad. It's really worth visiting to get a feel for the spiritual/religious side of modern Turkey which can often be forgotten about on a short visit to Istanbul. If you end up there during Ramadan (just started), it's a great place to go to break the fast in the evening as the shrine area is surrounded by street-side restaurants. From Eyüp you can take a cable car up the hill to the Pierre Loti cafe for views down the Golden Horn over the entire city.
Close to Eyüp a little way east along the Golden Horn you see the start of the old Byzantine Walls of the city which stretch all the way south to the Sea of Marmara. At the north end there are some Sufi shrines dedicated to more minor Islamic figures, these are worth checking out if you've the time and interest. You can then follow the walls all the way south, I spent a whole day doing this once, it's a long walk especially in the sun. There are stretches where you can actually walk along the top of the walls (with great views), but mostly you have to follow along the east or west side, through loads of traditional neighborhoods you'd never visit otherwise. There are some sites to see along the walk, such as the Chora Church, the new Panoramic 1453 Museum dedicated to the Ottoman conquest, and the old pigeon market.
Other neighborhoods I enjoyed wandering around were Fatih, (a conservative area west of the Grand Bazaar/University area where you can regularly stumble on to some ancient building which would be the number one tourist attraction in most other cities but in Istanbul barely gets any attention; Ortakoy, a little bit up the Bosporus by the massive Bosporus Bridge, there are some good restaurants and bars in the area around the square here; Beşiktaş which is a little further south of Ortakoy so closer to the centre, it seemed a bit busier then Ortakoy with loads more bars and cheaper shopping areas; and Kadıköy over on the little-visited Asian side of the Bosporus, there is a really vibrant, busy buzz around the main area back from the old Hadarpasa Train station and port area.
Then there are the day trips by boat - you can go all the way up the Bosporus to the Black Sea on a day trip, or over to the Princes Islands. Both are well worth your time. And I'd also recommend the military museum north of Taksim, they put on a little show every day I seem to remember with a band doing old marching tunes.