Early atmospheric Watt steam engine - how does the steam move from the cylinder to the condenser? Below is a picture of one of the earliest designs of Watt's steam engine.
The basic principle of operation is this:


*

*The weight which is attached to the beam $E$ pulls the piston $P$ up, sucking steam from the boiler into the cylinder $B$.

*In the condensor $C$, the steam is cooled and condenses to water. This greatly lowers the pressure creating a pressure difference between the atmosphere and inside of the condenser and cylinder. So the piston is pushed down, doing work.


My question is, what makes the steam move from the cylinder to the condenser? 

 A: The valve marked V' is opened.  The process is something like the following.
To start the engine open V and V' and allow both chambers to fill with steam (this is just a bootstrap you heed to get air out of it).  Now close both valves and follow the cycle below:


*

*spray water into the condenser which will cool the steam in it and form a partial vacuum (both valves are now closed, the cylinder is full of steam and the condenser is a partial vacuum);

*open V' (V is still closed) and steam from the cylinder rushes into the condenser, with the piston then being pushed down by atmospheric pressure until it almost reaches the bottom of the cylinder;

*close V' and open V, allowing steam into the cylinder, then close V (both valves are now closed);

*either after the last step, or while it is going on (they can be overlapped) go to (1).


The trick that is being done here is that the cylinder stays hot all the time: only the condenser is cooled.  This means you don't have to repeatedly heat the walls of the cylinder after spraying them with cold water which a Newcomen engine did.  This in turn means you don't waste a great lot of the steam you let in to the cylinder condensing on the walls to heat them.
You do have to repeatedly cool the condenser's walls, but remember that cooling water is essentially free: these things ran in environments where there was copious cold water, not least the water they were pumping.  Cooling water is cheap, but steam isn't.
A: tfb's answer correctly describes the working cycle - just wanted to capture some of the extra questions in the comments.
Ideally you want to fill the cylinder with steam to lift it and then have the steam disappear creating a vacuum to pull the cylinder down. In Newcomen's original engine this was done by spraying cold water into the cylinder, cooling the steam and creating a partial vacuum - air pressure on the top of the cylinder than pushes it down. (Incidentally this was the power stroke because they wanted the other end of the beam to go up to lift water out of the mine.) 
The problem with this is that you cool the cylinder walls on each cycle, so the fresh hot steam for the next stroke immediately starts to condense on the cold walls and it isn't until the walls have heated above boiling point that any steam can start to lift the cylinder.
Watt's breakthrough was the idea that since steam is a fluid you can remove it from the cylinder by connecting it to a separate vacuum. After the steam has filled the second tank (the condensor) you close the valve and spray cold water into the condensor, condensing the steam and creating the vacuum ready for the next stroke. (You also have to occasionally pump out the water from the condensed steam.)
By doing this you keep the cylinder hot and the condensor cold so you don't waste energy and ultimately fuel.
A: Here is a good animation which shows the process in question: https://www.youtube.com/watch?v=9gWV9gLuslw&feature=youtu.be&t=355
In the same video, there are also good animations of Newcomen's engine (https://www.youtube.com/watch?v=9gWV9gLuslw&feature=youtu.be&t=236) and Watt's second innovation that the piston can do work while moving in both directions (https://www.youtube.com/watch?v=9gWV9gLuslw&feature=youtu.be&t=489).
