Remember that the idea of the game is that every player is a time traveller trying to manipulate world events to create an alternate timeline, so that they can go home. To that end, at the start of the game everyone randomly receives an ID card to tell them who they are. This ID is kept secret from the other players.
This is a small sampling of the number of ID's that are actually present in the game. Each card provides some back story for who you are, and a list of three years. If a year listed is black, it represents a linchpin on the timeline. That linchpin needs to be in it's beginning untampered-with-by-time-travellers state. If a year is red, that indicates that you need to create a paradox on that year and patch it.
For example, to get back to his own timeline for Yuri, he needs to make sure that russia is the first country to get into space, and that russia doesn't collapse in 1991. In order to do this, he will need to make sure that JFK doesn't get assassinated (The US pressed on to be the first nation in space to honour the wishes of their dead president) and then patch the timeline.
To make sure that the USSR doesn't collapse in 1991, he must make sure that president Regan gets assassinated, so that he can't go on to befriend russia and end the cold war.
If Yuri can change the proper key events and then apply the proper patches, he will have restored the timeline for himself. He can then go home, which means he wins the game.
