Before the October Revolution, Russia was not much different from other countries regarding Christmas traditions. The only important bit of local color was the day of Christmas: the Russian Orthodox Church still observes the Julian Calendar, so their Christmas falls on January 7 for Gregorians. The rest was mostly the same: Christmas trees with garlands and decorations, the old grandfatherly gift-giver who was identified as St.Nicholas (Santa Claus). The New Year was a separate secular holiday estabilished by Peter the Great, and it was not yet intertwined with Christmas.
However, everything changed with the Revolution. At first, the Bolsheviks tried to suppress the winter holiday traditions entirely. Christmas was banned as part of general antireligious propaganda as a "bourgeois clerical tradition", and for the New Year the Bolsheviks simply didn't care. Things changed in 1935, when the Soviet authorities decided to strip the Christmas imagery from everything religious and reattach it to the New Year. Thus the current tradition was born.
Instead of Santa, a new Suspiciously Similar Substitute was created, named Ded Moroz (Grandfather Frost). He was loosely based on an old Slavic nature spirit Morozko (or Jack Frost), who didn't have any association with holidays and was "responsible" for cold and frost; the new Ded Moroz, however, was explicitly a Santa-like figure. A sidekick was also invented for Ded Moroz, his supposed grand-daughter named Snegurochka (Snow Maiden); likewise, she was loosely based on Russian Mythology and Tales but previously didn't have any connection with either Christmas or New Year.
After The Great Politics Mess-Up, to which the Soviet Union was formally dissolved on the day after the Gregorian Christmas of 1991, the little Christmas mess-up happened. Everyone was used to the Soviet New Year, and no one really knew what Christmas is supposed to be like. So it became a mostly quiet holiday for the religious, and every festive tradition remained attached to the New Year.
New Year in Russia has an equivalent of Christmas trees. A New Year tree is always called a fir tree (yolka) even when another conifer trees is used (like a pine). Kids believe that Ded Moroz (the Russian Santa; literally "Grandfather Frost") puts presents under the New Year tree on the Januray 1 night, when they are asleep.
Ded Moroz is usually accompanied by his granddaughter, Snegurochka (Snow Maiden). He carries a staff and a bag of presents. His garments are blue, sometimes red or in some rare cases golden. He travels in a sleigh, carried by trio of white horses. Some parents hire Ded Moroz impersonators to deliver gifts to their children personally, in exchange for a small poem recited by heart, usually New Year themed. Every children's New Year festival, usually carried out in schools and kindergartens, also has a Ded Moroz and Snegurochka. There's also a single official Ded Moroz impersonator who anonymously lives in Veliky Ustyug, Vologda Oblast and is supposed to represent the real Ded Moroz.
There's a number of peculiar traditions beyond that.
- Winter Festival Seasons: note the plural. There is the official holiday season, which lasts a week from the New Year to Orthodox Christmas. It is a universal holiday week, applying to school, university and work; only shops and emergency services work as normal, and the citizens sit at home and continue celebrating. There's also the unofficial holiday season, which is more than twice longer; it starts with the Western Christmas and ends with the "Old New Year" (see below). The Russian comedian Michail Zadornov said once, that if some American calls a Russian company near New Year, he may hear a recommendation to call again in April: even after the unofficial season is over, many people cannot get back into working mood quickly.
- New Year Supper: this is a big deal meal that is eaten during the New Year night. Usually it is had by the whole family, gathered in front of the TV. The stock dishes are Olivie salad (known as the Russian salad worldwide) and another kind of salad called "herring under a coat". Much alcohol is had, usually in the form of champagne or locally produced faux-champagne; it is customary to open the bottles at 0.00. It usually continues from 23.00 to 2.00 or 3.00. The New Year supper is a popular subject of Russian jokes, which typically make fun of the perceived tradition to drink so much champagne and vodka on December 31 that it's impossible to get up the next day — and when someone does, Hilarity Ensues.
- Presidential Address: just before 0:00, it is customary for families to gather in front of their TV sets and listen to the president's public speech congratulating citizens with New Year, and listen to the national anthem. So far, there were two seriously abnormal speeches first on December 31, 1991, read by the famous standup comedian Mikhail Zadornov: in the chaos of Soviet Union dissolution few days before no one up there thought about the traditional speech, and Zadornov just happened to be the most universally recognized figure that's been in studio at the time, and the second was on December 31, 1999, when Boris Yeltsin repeated his surprise resignation announcement from earlier that day, followed immediately by a more typical New Year speech by Vladimir Putin, then acting president.
- New Year Movies and TV: several movies about the New Year were made in the Soviet Union, and it is customary to show them on TV every New Year. Two most famous are The Carnival Night and Irony of Fate. There are also numerous musical TV shows made for the New Year, generally about local pop starlets having fun.
- Old New Year: celebrated by those who haven't had enough of New Year and Christmas on January 14, which corresponds to New Year's Day in the Julian calendar. Unlike New Year and Christmas, Old New Year is not an official holiday. The wildest party animals and the biggest alcoholics will start "celebrating" on the Gregorian Christmas and continue all the way until the Julian New Year.
- Fall semester exams in higher education. Traditionally, they start somewhere close to end of December with series of 'зачет's, before New Year and continues with proper exams (usually, but not always, after New Year holyday season). The problem is, that students of higher education are prone to long alco-maraphones and it so happens that the greatest holyday falls just in between exams.