You can create your own typing contest with fastfingers.net, just create or login to your account, select the typing language you wish to type in and you can create a typing championship that will last for 24 hours. The test works the same as the typing test, it runs for one minute and the faster you type, the higher your score and therefore the higher you will rank on the tournament highscore list. There is one important difference though: Once you created the competition the words will stay the same, so the more often you do the competition, the better your score will get.
You can repeat the competition as often as you like, only your best result will count, so if one of your friends just beat your amazin result, try again and take back the first place. After the 24-hour period is over the typist who has the best result will win the competition, this will also grant an achievement and is shown on your profile page.
After creating the speedtest competition you should invite your friends and fight for the victory or you can just create a public contest and compete against everyone on the site. If you just want to join a tournament you should take a look at all the currently running typing competitions.