When an American thinks of a party they think of a bunch of people getting together at a home, restaurant, or other large location to listen to music, socialize, and probably drink to some extent. There is a very superficial feel to all of this.
In Russia, a party is a much more significant event. It is a big deal to get invited to someone's home seeing as they tend to have a close network of family and friends that they usually associate with on such an intimate basis. At a Russian party there is usually a large table filled with food. Small appetizers, meats, cheeses, bread, 'pies,' potatoes, and pickled foods are common along with wines and vodka, the national drink. American 'party snacks' usually include pizza, potato chips, pretzels, and other finger foods along with sodas and alcohol to some extent.
The meals tend to be very long and consist of many toasts (which equate to many drinks). Everyone will drink unless they have good reason not to. If your drink is not finished after the toast it is impolite to put it down until it is.
In the U.S. it is common to have liquid chasers after drinking vodka, such as juice or soda, or making mixed drinks with vodka, in Russia it is not. Chasers are usually small pieces of black bread or another appetizer sized bite and mixed drinks are uncommon. These parties tend to las for a very long time and involve stories, jokes, music, songs, and general enjoyment of one's company. In the end, everyone is usually quite happy. However, it is considered in bad taste to become visibly intoxicated off of vodka
In general, Russian parties are much more intimate than parties in the US and involve more toasts (alcohol), more varied food, and probably a much more fun experience!