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. Formerly, they not only served Moscow–Saint Petersburg but also Moscow–Nizhnyi Novgorod, but they were phased out of the latter in favour of Talgo trains in 2015.
You can use the site of Russian Railways to search for departure times and travel times. Look for the САПСАН 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).