You can either sign up for a free remotely hosted board at somewhere like fastboards.com (never used them, just know the name!) or you can install a board on your own webspace. This is far prefereable as you will get far more control. You can get a free board such as http://www.phpbb.com and put it on free webspace (http://www.free-webhosting.info) that supports PHP and MySQL.
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While David is right in saying that installing a message board on your own webspace will give you more control, I recommend that you only do it if you have paid webspace. Free remotely hosted message boards tend to be far more reliable than free webspace providers offering PHP and MySQL.
Also if you decide to use a board on your own webspace you can check the free boards on this page http://www.opensourcecms.com/
You can even test the online demos that they have installed