My toshiba a200 laptop gets heated when playing a game and eventually shuts down.
Should I buy an external fan for?
Would it make a big difference?
If so, which kind of fan is a good one?

You can get a laptop heatsink that will raise it from the surface a bit and also better the airflow below it. This does help, but does not always solve this exact problem as heat is generated by the components inside.

You can also clean the fans to make sure they are not being slowed down by dust.

Dear Champion, As Joshua says, the Laptop support adapter is ideal for increasing the air flow. Cleaning out the fan ducts with a suitable adapter made and fitted to some hose type vacuum cleaners (set to slow) will also help ....if you are mechanically minded

However the main cause of most problems of this nature are using the Laptop lying on soft furnishing ie in Bed, Carpets, and any surface which has a cloth/wool linted surface, which allows dust and wool/cotton fibres to clog the fan/fans and restrict the air flow.
This problem I have solved for dozens of members on this site over many years has prooven with feed back in most cases to be a useful remedy.
Useful manual link to the Toshiba
http://forums.vnunet.com/thread.jspa?messageID=1098039&tstart=0#1098039

Best wishes to you both
Vic UK & Germany

Thank you so much guys.