The first real hostel was opened in Moscow in 2005 - owned and run by a westerner - Godzilla's hostel has the central location and the right atmosphere for socializing. Occupying three apartments in a pre-revolutionary residential building near metro Tsvetnoy Boulevard (and Tverskaya), the hostel has eight dorm rooms and two private doubles. The rooms have a warm atmosphere, nice design, and the windows overlook a quiet street and a courtyard. Some rooms are not as nice as the others though, so if you have a chance, get the rooms with fewer beds.
A problem is that the hostel is often completely booked.
Another problem might be that the rooms have lots of beds, and that there's only three bathrooms to share between everybody, so prepare to queue in the mornings... Also, most of the beds are two-level (see photo), so if you're a couple make sure you ask for a double bed.
The prices are standard for Moscow: $16 for a bed in a 10-bed dorm, $28 per bed in a dorm room (sleeps from 4 to 8 people), and $70 for a private room (single, double or twin). This latter option makes it a very attractive place if you want some affordable accommodation in the center of the city and still care about privacy.
All the standard facilities are offered: two kitchens, lockers, luggage room, safe, washing machine. There are no lockouts, no curfew. The internet access is free for guests and the English-speaking staff will help you to find your way around the city.
The hostel provides visa support through the company called Get Russian, which we don't really like, so it's up to you whether to use them or not.
Finally, Godzillas also has a hostel in Suzdal, a small picturesque town a couple of hours away from Moscow.