There are three types of trains in Russia - high-speed trains, luxury overnight trains and regular trains. If you are traveling between one of Russia's three major cities - Moscow, Saint Petersburg or Nizhny Novgorod - you can take a high-speed Sapsan train or an overnight train. If you value your time, take the high-speed train. If you want a nice experience and comfort, take one of luxury overnight sleepers. If you are looking for a bargain, take a regular overnight train.
Express trains are European-standard high-speed inter-city trains, capable of typical ground speeds of 250 kph (or 155 mph). They currently run between Moscow, St. Petersburg, Helsinki, and Nizhny Novgorod. These trains are called "Sapsan" within Russia, or "Allegro" on the Helsinki – St. Petersburg route.
The second type of trains are regular trains. Regular trains operate on just about any route between cities connected to each other by railroad, including those served by the high-speed rail trains (Helsinki, St. Petersburg, Moscow, and Nizhny Novgorod). However, these are usually reserved for overnight routes, and they offer beds for seating rather than individual seats.
Luxury overnight trains always have names. For example, the overnight trains from Moscow to St. Petersburg that depart at midnight and arrive at 8am the following day are called “the Grand Express” and “the Red Arrow”. These trains are part of Russian history and offer an experience in itself. Beds are comfortable, noise isolation is outstanding, and the onboard restaurants serve quality food and drink. In addition to the 1st and 2nd class cabins, these trains also have VIP cabins with showers.
Another important point that you should consider is that Russia's capitals of Moscow and Saint Petersburg have more than one rail station. They are all located in the city center, of course, but you should pay careful attention to where your train departs from.