If the chain/cog no longer drives the wheel, can this be fixed? Suddenly (without anything sounding like it was broken) the chain no longer turns the back wheel. The cogged part of the wheel is turned by the chain but this doesn't turn the wheel.

Is this something I can fix? I can't see a problem but I haven't started to try to take anything apart yet.

Please note this is a child's bike so there isn't much value in the bike or its parts. Just wondering whether or not to send this bike to the skip.
There are four main possibilities, depending on the quality and age of bike.

The most likely scenario is that the wheel is a traditional kid's coaster brake wheel - one gear forward, pedal backwards to brake.  In that case it's likely that the clutch is slipping inside the hub. A replacement wheel is probably an easy find, even more so than replacement parts, since kids have a tendency to outgrow their bikes rapidly.  If you're feeling wrenchy however, you can take a look here at an extremely detailed teardown and explanation.  You won't need any particularly special tools other than thin wrenches, preferably real cone wrenches.

If the wheel is on a nicer bike with hand brakes you'll be looking at either a freewheel or freehub design.  If it's a freewheel then you've got two possible culprits: you've stripped all the threads off (unlikely) or the pawls are frozen or broken.  In the former case, you'll need to replace the whole wheel.  In the latter, you'll probably just want to replace the freewheel outright - repairing them is annoying.  Replacing it will need a special tool that matches the splines/notches on your freewheel. 

The freehub design is similar, except the pawls are in the hub body itself.  You could be lucky and simply need to tighten the freehub assembly, but once again it's probably frozen or broken pawls.  Reparable, but a bit tricky and once again needing some specialized tools.

The last option is that it's some kind of internal gearhub. If it's broken they can be fun to take apart, but you're not likely to get it working again.  If you've got a local bike co-op they might want the donation.