Skip has covered most of what needs looking at, so I thought a 'plan of action' might be helpful.
i) - Make sure that the string is not slipping on the winding post !!!
This will show up if the winding post is turning, but the string is not getting any tighter. Most players have their own patent method of attaching strings to the winding post to prevent this happening. Just poking the string through the hole and giving it a couple of turns may not be enough to stop it pulling back through when it's tightened.
ii) - You will have seen from (i) whether the winding post is turning or not, if it isn't turning then the slippage is somewhere else.
So, take off the tuner by removing the nut from the inside of the pegbox, the tuner should just pull out.
The tuner is made up of sections, very much like the transmission in a car.
The winding post is the prop shaft that turns the wheels.
The barrel section is a single speed gear box.
The button is the engine.
The fibre washer is the clutch plate. You can only get at this by taking off the button.
iii) - Examine the underside of the barrel, and make sure the small pin is in place. This digs into the outside of the pegbox and stops the barrel rotating.
iv) - Take off the button. Hold the barrel between your fingers and turn the button shaft, does the winding post also turn. If it doesn't then most likely the gears are stripped and it needs replacing.
v) - Examine the fibre washer and look for any parts that have become shiny with wear.
vi) - You might not need to do this, depending on (v), but do it anyway.
Put a piece of fine sandpaper on a flat surface and give both sides of the washer a gentle rub over, just enough to rough up the surface a bit.
Most unlikely, but take a look at the bottom of the button itself for any signs of wear. If you do any sanding on this part be very careful to keep everything flat.
vii) - Put the tuner back on, ensuring that when you tighten up the nut the pin digs into the wood and the barrel is tight and flush.