I have 2008 Yamaha Scorpio 225, and the chain is looking a bit loose. I've had a good look around, and I really can't find a way to tighten it.
Does anyone have any idea where to look? Is there a common system in use on motorbikes?
If the pictures I'm seeing of your bike are correct, you have the adjusters built into the swing arm. The proceedure should be in the owners/service manual for the bike, but here is a brief overview. I cannot stress enough that you should disregard the procedure below and reference the owners/service manual instead.
On that note, you basically loosen the axle enough that you can push the wheel forward (usually hit the tire with a sledge to make sure it moves forward). Then find the two adjusters (nuts and/or bolts) which are hanging out the back of the swing arm. They should have a locking nut, which you will have to break loose. You'll then screw the adjusters out, which will allow the wheel to slide backwards by grabbing the wheel and pulling.
Once the wheel is far enough back to properly tension the chain (again, see the owners manual), check the alignment. There should be notches on the swing arm to help with this bit. Once the axle is square and the chain is correctly tensioned (not too tight! See the owners/service manual!), lock down the adjusters and tighten back down the axle to the correct torque specs (manual, again. It might be higher than you think, my Hayabusa is specced to 130 ft/lbs).
Well, usually one would go about looking in the service manual... which if you plan on doing your own maintenance is essential and well worth the money. Typically on a motorcycle there are adjustment screws on the back of the swing arm that you screw one way to tighten and the other way to loosen. You need to loosen a bunch of stuff first (axle, adjuster screw lock nuts, torque link - if applicable), then screw adjustment screws, then check the wheel alignment, then tighten everything back up.