This was the Great Comet of 1811, which could still be seen in the night sky in 1812. First discovered by Honoré Flaugergues, it was visible to the naked eye for 260 days, making it the second most visible comet after Hale-Bopp in 1997. Its association with Napoleon's invasion of Russia led to it being known as 'Napoleon's Comet'. Its scientific designation is the slightly less catchy 'C/1811 F1'.
This comet's appearance marks a slight problem in Tolstoy's timeline: he accidentally left out 1811. Natasha's engagement to Prince Andrei spans 1810, up until Anatole's appearance at the 'end of January' 1811. The crisis happens over about ten days, taking them to mid-February by the time Pierre becomes involved. Pierre notices the appearance of the Great Comet in the sky above Moscow, pinning him to February 1812, although he has just come from comforting Natasha in 1811, earlier that same evening.