SoHo is a really nice area with lots of good restaurants and cafes, and some of the best shopping in the city.
Lower East Side & East Village are popular neighborhoods, especially among younger people. This part of the city has an excellent selection of restaurants and bars, and with a local, authentic feel, which is otherwise hard to find in Manhattan.
Meatpacking District/Chelsea is particularly popular among European tourists who want to experience NYC nightlife as they know it from the reality shows. There are some really nice (and expensive) hotels in the area, plus you have Chelsea Market within walking distance.
Midtown is mostly for tourists, and except for Times Square, the area is rather dead at night, when the office workers have gone home. Though if it is the first time you visit New York, and if you are only there for a short while, you might find that the convenience of being close to many of the city’s main attraction outweighs the cons of staying right in the tourist hotspot. You also have Koreatown and a great selection of Korean restaurants nearby.
Financial District is not the most interesting area to stay, and it is hard to find good restaurants and bars. Though if you are visiting New York City on a weekend, when all the bankers are out of the city, you can often find good deals on hotel rooms. Furthermore, you are only a few subway stops away from more interesting neighborhoods in Lower Manhattan.