When you go to Russia, you'll always have to fill this little form. The so-called migration card. It's just a tiny piece of paper, not really a "card".
The migration card is a double-form where you have to fill in some of your basic data. It's then stamped by the border guards and split in half. They keep one half, you get the other and keep it until you leave the country again. From what I've heard, one of the more troublesome things you can encounter is losing your migration card while in Russia.
There are some tricky things to know about the migration card. First: If you enter Russia through Belarus, you'll get only one migration card for both countries. That bothered me a lot on my last trip, because I only entered the data for Belarus, that means a stay of one day. Turned out this wasn't a big issue. When leaving Russia the border guards had a more closer look on the documents and obviously noticed the misinformation, but they didn't seem to care that much.
There seem to be similar issues with other countries. I even once got a migration card at the Ukrainian border, where I didn't want to enter Russia at all. It just happened that I was sitting in a train that was going to Russia after crossing the Ukraine. So if you ever happen to enter Russia through another country, although you don't necessarily get the migration card at the Russian border, you should enter the data for your stay in Russia.
Another thing I learned this time: There is a field for the Visa number. Now I obviously had two Visa - one for Belarus and one for Russia. I filed the number of the Russian one, because after all, I wanted to go to Russia and only cross Belarus on a train. Turned out I was wrong and the Belarus border guards complained. Admittedly, it can be a bit frightening if a border guard tells you something is wrong, especially if you don't understand their language. But after all, he just filled in the correct Visa number himself and everything was okay.
Kazakhstan also has a migration card, but it is not split in half and no similar pitfalls seem to occur there.