Found this online (I hope to add or modify more information later):
The Russian Bear is a national personification for Russia, used in cartoons and articles at least since the 19th century, and relating alike to Tsarist Russia, the Soviet Union and the present post-Soviet Russia.
The bear image was, however, on various occasions (especially in the 20th century) also taken up by Russians themselves. Having the teddy bear "Misha" as the mascot of the 1980 Moscow Olympic Games - boycotted by numerous countries due to the invasion of Afghanistan - was evidently intended to counter the "big and brutal Russian Bear" image with a small, cuddly bear.
After the collapse of the Soviet Union, there was some support in the Russian Parliament for having a bear as the new Russian coat of arms - with the proposers pointing out that "Russia is anyway identified in the world with the Bear" - though eventually it was the Tsarist coat of arms of the Double-headed eagle which was restored.
Later, the bear was taken up as the symbol of the United Russia Party, which at present dominates political life in Russia. Coincidentally, the family name of Dmitry Medvedev, the Russian president elected from that party in 2008, is the genitive of медведь, meaning "bear".
So now Russians can nod their heads in agreement when foreigners ask: "Are there bears on Red Square?"