All the players then have a certain amount of time, specified in the game parameters, to study the situation and determine a set of movement and combat orders to issue their worlds and fleets for next turn. Note that all the players work on their turns at the same time, independently of each other (though usually there is much negotiation that happens between players while they work out their turns).
RSW games are commonly played at the rate of about one turn per week, but any turn rate is possible; some people enjoy the excitement of playing games at the rate of one turn per day or faster. Generally, longer turn times allow for greater amounts of negotiation and diplomacy between players, so you can choose the turn rate that suits your playing style. It is possible to play a blitz game that rolls over a new turn every few minutes; games of this nature will typically involve very little diplomacy, and can be finished within an hour or two depending on the turn rate.
Out of kindness for your fellow RSW players, please do not sign up for a game unless you are ready to make the commitment to play it to completion. If you get halfway through a game and decide you need to drop out, it spoils the game for everyone.
The end of the game is specified in the game parameters. Usually the game is over when one of the players reaches a certain score. At that time, additional points are awarded based on various end-of-game bonuses, and the player with the highest score is declared the winner. The target score is determined at random when the game is created, and is not revealed to any of the players until the game is over. It will generally be somewhere in the range of 6,000 to 9,000 points.