Fast train between Moscow and Saint Petersburg?

As Volodymyr mentioned, the train is called Sapsan, Russian for peregrine falcon. It is a Siemens-built high-speed train based on the ICE 3 running in Germany and Siemens calls it Velaro RUS. Some also serve the Moscow–Nizhnyi Novgorod relation.

You can use the site of Russian Railways to search for departure times and travel times. Look for the ?????? text below the train number to identify them. Note that the departure times are not spread out across the day. Rather, there is a ‘morning cluster’, an ‘evening cluster’ and on some days, a ‘lunchtime cluster’. The journey time varies between 3:40 h and 4:05 h, probably depending on the number of stops on the way (didn’t check to confirm the hypothesis).

To board, be there timely at Moscow Leningradsky station, be prepared to have your luggage screened as if boarding a flight, and then present passport and ticket to the conductors when boarding as is the norm for Russian long-distance trains.

The train ride itself was very pleasant (I travelled with the cheapest option possible in late summer 2015) and a small meal was included (a sandwich or so if I remember correctly).

There is also Nevskij Express.

It is more like classic Russian/European trains, cars of compartments with actual doors and 6 seats per compartment. I would argue there's more personal space. Ticket price is usually comparable to Sapsan. Departure time is not very convenient (it's a day train) but it's in demand. It takes 4:05h without any stops.