How do I fix a ceiling fan that won't turn off?
Depends on how it is wired. The single switch will control the fan and light simultaneously. If you want the light off, then you pull the chain for the light. The nice part about having the light on a different switch is you can simply go into a room without having to turn on the fan every time and have a nice bright fixture in the room. The downside is that you have to hit the switch every time since the pull chain kind of loses its purpose and is now redundant. 

Now, whether you have three speeds or not is up to the manufacturer of the fan. Most fans made recently have three speeds. We have to know the make/model to find out for sure. Doubt it is broken though.