Warsaw has three stations for long-distance trains:
Dworzec Centralny or Warszawa Centralna (Warsaw Central) ul. Emilii Plater
Dworzec Wschodni or Warszawa Wschodnia (Warsaw East) between ul. Kijowska and ul. Lubelska (on the right bank)
Dworzec Zachodni or Warszawa Zachodnia (Warsaw West) Al. Jerozolimskie near Rondo Zeslanców Syberyjskich
Unless you really know what you're doing, the best option is Dworzec Centralny (Warszawa Centralna) station, as it has best connections with all the places in the city. All long-distance trains pass through this station and all stop there. It is the only long-distance station underground. It isn't the last station on the route!
Trains running eastwards start at Warszawa Zachodnia, stop at Warszawa Centralna and then at Warszawa Wschodnia, while trains heading westwards make the same trip in the opposite direction (except that they don't always stop at Warszawa Zachodnia). The same is true for arriving trains. Tourists often find it confusing that the main train station (Warszawa Centralna) is not the last station on the route.
The Berlin-Warszawa Express runs from Berlin to Warsaw daily and is quite inexpensive if booked in advance. Students get discounts as well. It's a fairly comfortable six hour trip. Schedule are available on Intercity's website and tickets can be booked through Polrail Service.
One train per day runs to Kaunas and Vilnius in Lithuania, with a change of trains near the border.
There're also trains to Russia - Moscow, Saint Petersburg and even direct cars to Saratov or Irkutsk, Belarus - Minsk, Ukraine - Kiev and direct cars to Simferopol or even Astana in Kazakhstan. There are many direct connections to west-european cities too.
In domestic trains, reservation is obligatory only on EC, EIC or Ex trains. In the most popular TLK (cheapest, sometimes only a little slower than expensive EIC) seat reservation is possible (but not obligatory!) only in 1st class.
There are also some IR (InterRegio) trains, operated by Przewozy Regionalne. These are the cheapest trains in Poland, but often not very comfortable and slow. Only IR Warsaw-Szczecin-Warsaw (called "Mewa") and Warsaw-Rzeszow-Warsaw (called "WOSP") are highly recommended - they're equipped with air-conditioning, CCTV, power sockets for every seat and free wi-fi connection, although the price is extremely low. Seat reservation is possible for as little as 3zl (~0,75?), but only when purchasing internet ticket on website.
If you don't have a reservation, you may get a better seat by boarding the train at its point or origination. During peak season, you may not be able to get a seat at all.

Suburban trains
There is no central station for suburban trains, but the most important one is Warszawa Sródmiescie (in close proximity of Warszawa Centralna and Metro Centrum). Some destinations you can reach from there are Grodzisk Mazowiecki, Lowicz, Milanówek, Minsk Mazowiecki, Otwock, Pilawa, Pruszków, Radom, Siedlce, Skierniewice, Sochaczew, Sulejówek, Terespol, Tluszcz, Wolomin, Zyrardów.
Suburban trains going north depart from Warszawa Gdanska (Metro Dworzec Gdanski). From there, you can go to destinations like Ciechanów, Dzialdowo, Mlawa and Nasielsk.
Similarly, suburban trains going east depart from Warszawa Wilenska (ul. Targowa near Al. Solidarnosci, also a shopping mall there). Some example destinations include Malkinia, Tluszcz, Wolomin.