Central Amsterdam is the area running from Centraal Station along the Damrak to Dam Square. It is the most touristy part of Amsterdam, aside from the infamous Red Light District and the Leidseplein, and the first to meet travelers arriving by train. Here, hotels are expensive and shops attempt to wring the last dollar out of a tourist's wallet. It is bright, noisy and active. It is still possible, however, to happen upon some finds in the area, and it is a real knock-out at night, if you can block out all the noise.

Red Light District lies just to the east of Central Amsterdam, and is known for the prostitutes who legally ply their trade there. People walk the streets at all hours of the day and night, gawking at the women in the windows. It is not necessarily dangerous, but visitors are advised to stay on their guard, especially late at night,(do not even talk to the drug dealers, get your stuff in a coffeeshop or smartshop) as people will approach you on the street trying to illegally sell drugs. The Red Light District is bordered by the small Amsterdam "Chinatown" along the Zeedijk.

Leidseplein is one of the best places to go to enjoy Amsterdam's nightlife. The square itself and the surrounding streets are the most touristy in Amsterdam, and a collection of bars and nightclubs assure that there is always something going on. A great number of restaurants of all different ethnicities can be found in the side streets that lead away from the main square.

Rembrandtplein is similar to the Leidseplein and is the second most important place for nightlife. The Rembrandtplein is more popular with locals, but also has fewer bars and restaurants than the Leidseplein. Rembrandtplein has a few restaurants, mostly bars or pubs with many more within steps. It is extremely loud around the square during popular times. Club Smokey is loud during winter and spring. The square has a small park and borders the gay and lesbian district of the Reguliersdwarsstraat. If you want to smoke and drink Rembrandtplein is the place for you. This is not a family area of Amsterdam. The Irish Pub just below Hotel Atlanta serves up a fantastic full Irish breakfast.

Jordaan is the area west of Central Amsterdam is a haven where quiet streets wind along sleepy canals. Historically a working-class neighborhood, the Jordaan seems untouched by the more seedy aspects of Amsterdam. It is a nice place to stroll or ride a bike and get lost among the gardens, markets and pleasant boutiques that line the streets.  

The Plantagebuurt is the 2nd museumquarter in Amsterdam, next to Museumplein. In this lively and green area you can find the Museum of the Tropics, the Jewish Historical Museum, a beautiful botanical garden and the Netherlands’ oldest zoo. Also you can visit the Hermitage, a branch of the Hermitage in St. Petersburg. But take some time to walk around, enjoy the neighbourhood and have a picknick in Wertheimpark or Oosterpark. Take a day trip here and discover the numerous museums, botanical gardens, television studios, tree-lined streets and the beautiful terraces along the water.
A lot of information on the neighbourhood can be found here: De Plantage Amsterdam 

De Pijp is a vibrant neighbourhood to the south of the Heineken Experience which encompasses the Albert Cuyp market. The only area of Amsterdam to have more bars and restaurants per sq.m is the Jordaan. In fact there are three Michelin starred eateries (Ciel Bleu has 2 stars) in De Pijp along with many other very good bars etc. There are not that many hotels in the area with the exception of the Okura (5 star) and the Savoy but there are apartments for rent and there are good transport links to the rest of the city. If you want somewhere with a great local vibe then this is the area to stay or even visit. On a sunny day take a seat outside a bar on the Marie Heineken Plein (directly behind the Heineken experience) and watch the locals at play.