Istanbul is a very populous city. Below is a collection of some of the older and central districts/neighbourhoods.
FATIH (SULTANAHMET, EMINONU, BEYAZIT, …)
Fatih is the district that contains the historical centre of Istanbul inside the once impenetrable city walls. It has been the grounds for the capitals of the Ottoman, Eastern Roman/Byzantine and the Roman Empires in its long history for a continuous period of more than 1500 years. It is home to many of the most important sights including Haghia Sophia,Topkapı Palace, and the Grand Bazaar. There is a lot to see concentrated in this area, with many high quality museums, historical bazaars, historical libraries, specialist shopping and artisans streets, walls of the city, the terminus station of the Orient Express as well as a wealth of eateries and agencies for touristic activities.
BEYOĞLU (TAKSIM, CIHANGIR, …)
Beyoğlu (also known as Pera) is the prominent centre of arts, entertainment and night-life in Istanbul. It’s been the centre of Western influence in Istanbul in the 19th century when it was home to a Levantine population of European traders. Nowadays it is home to many night-clubs, bars, art exhibitions, trendy shops and historic malls, numerous mainstream and alternative cinemas, theatres, eateries, bookshops, cafés, museums, churches, synagogues and hotels.
Beşiktaş is an affluent cosmopolitan district which has gained prominence beginning from the late 19th century as the Ottoman palace moved here. The centre of the district also known as Beşiktaş is host to numerous historical sights (such as the Dolmabahçe Palaceand the Yıldız Palace), museums and eateries while retaining a residential feel. Beşiktaş as a district contains many interesting neighbourhoods:
Ortaköy is a lively neighbourhood which pulls crowds day and night. The centre of attraction is the Ortaköy square by the sea which has many tea-gardens, cafes, bars and an open-market (mostly ornaments) where you can try the speciality ‘filled potato’ (‘kumpir’) while going around. Ortaköy also has a very lively night-life with many bars scattered around the square as well as the mega-clubs by the Bosphorus.
Arnavutköy is a seaside neighbourhood with beautiful yalıs (historical seaside mansions) lining by the shore. Formerly a predominantly Greek and Jewish neighbourhood it retains part of that heritage especially in its architecture. It has restaurants with views of the Bosphorous, bars, and Greek style taverns.
Bebek is a charming affluent neighbourhood further north by the Bosphorus. A walk by the Bosphorus from Bebek village to the 15th century fortress Rumelihisarı takes you along various seaside tea-gardens, eateries and cafes going under the huge Fatih Sultan Mehmet Bridge.
Populated from mid 19th century, Şişli used to be the choice of neighbourhood of the wealthy and the carefree until the mid 20th century. Of particular mention is Nişantaşı which retained much of the glory of the previous days with a busy and upmarket high-street.
Kadıköy is a cultural centre and a cosmopolitan middle-class neighbourhood on the Asian side of İstanbul. It’s mostly residential and may be interesting for its selection of restaurants, bars, shops and artistic activity. Kadıköy centre has numerous well-kept budget hotels which are an easy ferry ride from the historical centre (Fatih/Sultanahmet).
Bağdat Avenue (Bağdat Caddesi)
Bağdat Avenue is a well-known high-street in the surrounding affluent area by the coast starting from Kadıköy. It has outlets for the common up market brands along with many cafes and restaurants.
Wikipedia has a category for various neighbourhoods of Istanbul.