No doubt, you may call home or anywhere you like from your hotel. However, you will have to pay significantly more in this case, rather then when using public phone.
There are public card-operated phones in the entrance halls of Moscow's Metro stations, which may connect you with someone abroad. The cards are available in cashier windows at the same Metro stations, the approximate price is 15 USD ( in rubles equivalent). The credit left on the card is shown on the digital display of the card-operated phone.
You may also call from one of the post offices, which are usually called "Post, Telegraph, Telephone", where operator connects you to required number. You may also buy another plastic card there and use preference tariffs even when calling from the hotel. Besides, there is a 24-hour discount for international phone connections on holidays and weekends and for the international night calls on weekdays.
Local calls are free, but that may change soon. Not all phones can make long-distance calls, but for those that do, dialling 8 will get you a long-distance line. Thus, to dial another region of Russia, dial 8, wait for the new dial tone, then dial the rest of your number.
For international dialing, dial 8, wait for the tone, then dial 10, then the country code and the rest of your number.
For example, to call the U.S. number (123)-456-7890 from Russia, you would dial 8-(wait for dial done)-10-1-123-456-7890