A Station allows you to use one, and only one, of the routes departing from that City but belonging to another player to complete your own Tickets.
To build a Station, simply drag a Train card from your hand onto the empty City of your choice.
The 1st Station you build will only cost you 1 card, while the 2nd and 3rd will cost you 2 and 3 Train cards of same color respectively.
Each unused Station is worth 4 points at the end of the game, so don't build them unless you really need them!