The PC game I will installing for you is known as Portal, one of valves most well known games. To begin the install you will click on the INSTALL button right underneath the name of the game in your library.
This box will come up, telling you how much space it will take up, how much space is available to use and a commonly pointless estimated download time, I say pointless because commonly most people with gaming PC's will be able to download a lot higher than 64 KB/s. There is also a drop down menu that allows you to choose where you want to install the game, but you may want to leave this alone.
The next screen you go to will have two options. Create start menu shortcut to Portal and Create desktop shortcut to Portal, these will allow you to access the game from two different places other than steam, including the start menu on windows PC's as well as a shortcut places on the desktop, these are optional, hence the appearance of tick boxes.
The next screen simply confirms the game is installing, you can now quit this box and go back the main steam UI.
Going to the download page on steam will show you what is currently downloading, how fast its downloading, and how much time is remaining, in my case I was downloading at 3.8 MB/s, which is pretty good, and the game will only take about 8 minutes to install!
This is an image of the game nearly finished installing.
One the game is complete, it will confirm this by saying the time completed, and how much was downloaded. From this point you can access the full game through your library, enjoy playing!