Suddenly the fan is coming on and staying on. Why, what to do?

Reset the SMC and you should be good to go.
http://support.apple.com/kb/ht3964

I agree a SMC reset will do the trick, on a MBP the correct process is:




*SMC RESET*

• Shut down the computer.
• Plug in the MagSafe power adapter to a power source, connecting it to the Mac if its not already connected.
• On the built-in keyboard, press the (left side) Shift-Control-Option keys and the power button at the same time.
• Release all the keys and the power button at the same time.
• Press the power button to turn on the computer.



Roger

Do you have any peripherals plugged in, if so please unplug them and re-try the SMC reset my instructions. If that doesn't do it then check to see if you a runaway process. You will need to open Activity Monitor (Applications - Utilities - Activity Monitor) and look to see if any processes are taking up a lot of % of CPU.

OK, there are several other things we can try. First do a PRAM reset:




•PRAM RESET•

• Shut down the computer.
• Locate the following keys on the keyboard: Command, Option, P, and R. You will need to hold these keys down simultaneously in step
• Turn on the computer.
• Press and hold the Command-Option-P-R keys. You must press this key combination before the gray screen appears.
• Hold the keys down until the computer restarts and you hear the startup sound for the second time.
• Release the keys.



If that doesn't do it try starting in Safe Mode.

And if that doesn't help run Apple Hardware test in Extended Mode to see if there are any error codes. If there are please post them and we can try to translate them.