Clearing your download cache can solve problems with games that won't download or start.
This process will not affect your currently installed games, but you will need to log into Steam afterwards.
From your Steam Client, open the Settings panel by selecting "Steam > Settings" from the top left client menu.
In the Settings panel, select the Downloads tab and you'll find the "Clear Download Cache" button at the bottom.
Then select "OK" to confirm and acknowledge that you'll need to login to Steam again.
If you're still having trouble with your downloads and any games that won't start, please contact Steam Support.