A Russian carriage or sleigh pulled by three horses that has become a revered symbol of Russia itself.

The troika was developed centuries ago to overcome the vast expanse of the Russian Empire, while allowing for the negotiation of difficult roads at great speed. A carriage is used in the summer and a sleigh in the winter, though because there is snow half the year in many parts of the country, the sleigh version is generally more popular even in the summer. The horses used are typically Orlov Trotters, which are said to be the best troika breed, and are arranged with the centre horse between the sleigh shafts and the outside horses guided by splinter bars attached to the sides of the sleigh. The centre horse moves in an extended forward trot, the side horses gallop smoothly with their heads bent outward, and they are all driven with four reins: two for the centre horse and one for each outer horse. The synchronized swinging motion of the troika has often been compared in literature to a bird in flight (see the longer passage quoted below).

As one would expect, troika are elaborately decorated with leather tassles, chains, brushes, rosettes, and stars, among other things. The duga, which is the semi-circular wooden bar that connects the main shafts over the centre horse (described in the shorter passage below as an "enormous wooden head-collar"), is often decorated in a light blue flower motif and tends to be very colourful, while bells traditionally hang from it and can be heard from a great distance. These carriage decorations, along with the horses that pull the carriage itself, are judged at Russian winter festivals, but of course, in the ever-popular troika races, only speed matters.

A first-hand account of riding in a troika from Frederick Burnaby's 1877 book, A Ride to Khiva:

This type of sleigh was murder to ride for anyone cursed with short legs. A trio of horses was harnessed to it, their coats white with pendent icicles and hoar-frost; as they ran, the driver stopped them from time to time and cleared the thick icicles from their nostrils and muzzles. The central horse ran in the shafts, his head fastened in an enormous wooden head-collar. This was bright with painted colors, but forced him to trot ceaselessly with his head held high, a brave sight yet surely taxing to the beast; meanwhile his companions on either side were harnessed by cord traces to splinter-bars attached to the sides of the sleigh. They galloped while he trotted, their necks arched in a direction opposite to that of the central horse. There was a jingling bell suspended from the peak of the central horse's collar, and the sight they made as he briskly trotted and they galloped at his sides, was very picturesque. A troika with a well-trained team could cross the Russian steppes at the spanking rate of twelve miles per hour.

The most celebrated troika-related passage in Russian literature is found in Nikolai Gogol's Dead Souls, which I think is where the troika's symbolism of Russia comes from (the footnote is my own):

For what Russian does not love to drive fast? Which of us does not at times yearn to give his horses their head, and to let them go, and to cry, "To the Devil with the world!"? At such moments a great force seems to uplift one as on wings; and one flies, and everything else flies, but contrariwise--both the verst¹ stones, and traders riding on the shafts of their wagons, and the forest with dark lines of spruce and fir amid which may be heard the axe of the woodcutter and the croaking of the raven. Yes, out of a dim, remote distance the road comes towards one, and while nothing save the sky and the light clouds through which the moon is cleaving her way seem halted, the brief glimpses wherein one can discern nothing clearly have in them a pervading touch of mystery. Ah, troika, troika, swift as a bird, who was it first invented you? Only among a hardy race of folk can you have come to birth--only in a land which, though poor and rough, lies spread over half the world, and spans versts the counting whereof would leave one with aching eyes. Nor are you a modishly-fashioned vehicle of the road--a thing of clamps and iron. Rather, you are a vehicle but shapen and fitted with the axe or chisel of some handy peasant of Yaroslav. Nor are you driven by a coachman clothed in German livery, but by a man bearded and mittened. See him as he mounts, and flourishes his whip, and breaks into a long-drawn song! Away like the wind go the horses, and the wheels, with their spokes, become transparent circles, and the road seems to quiver beneath them, and a pedestrian, with a cry of astonishment, halts to watch the vehicle as it flies, flies, flies on its way until it becomes lost on the ultimate horizon--a speck amid a cloud of dust!

And you, Russia of mine--are not you also speeding like a troika which nought can overtake? Is not the road smoking beneath your wheels, and the bridges thundering as you cross them, and everything being left in the rear, and the spectators, struck with the portent, halting to wonder whether you be not a thunderbolt launched from heaven? What does that awe-inspiring progress of yours foretell? What is the unknown force which lies within your mysterious steeds? Surely the winds themselves must abide in their manes, and every vein in their bodies be an ear stretched to catch the celestial message which bids them, with iron-girded breasts, and hooves which barely touch the earth as they gallop, fly forward on a mission of God? Whither, then, are you speeding, O Russia of mine? Whither? Answer me! But no answer comes--only the weird sound of your collar-bells. Rent into a thousand shreds, the air roars past you, for you are overtaking the whole world, and shall one day force all nations, all empires to stand aside, to give you way!

The word "troika" is generally used in Russian for groups of three, and it is also the title of a traditional Russian folk song (sung beautifully by the Red Army Choir).

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¹ A verst is a Russian measure of distance equivalent to just over a kilometre, or about two thirds of a mile.

REFERENCES:

http://www.horses.ru/horsemanship/main_troika.htm
http://www.iras.ucalgary.ca/~volk/sylvia/TroikaandTarantass.htm
http://ibiblio.org/gutenberg/etext97/dsols10.txt

The Troika is also a great Russian dance.
You dance it in groups of three in a circle (a star-like circle, each group of three as a line of the radius of the circle). The person in the middle of the three is the woman, the other two men.
The music is quite fast, so the outermost person has to "run" quite a lot, which is a good way to get rid of the surplus of energy in your body.
Basically, the dance is as follows:

   
    \  |  /
     \ | /
      \|/
   ---   ---
     / | \
    /  |  \
   /   |   \

each line of three  "---" is a troika of two men and one woman

- part 1. you dance/run with the three on your circle-line, counter clockwise.
- part 2. the outermost man dives under the arms of the two others, then the man in the center does the same.
- part 3. make a little circle with the tree and "spin" around.
- part 4. throw away your woman to the men in front of you, and pick up the one behind you.
the whole thing starts over again.