Step 1: Float ball
First off, lift the cistern lid and check that it is full of water. If it isn't, check the float ball, the big plastic ball in the cistern which controls the level of water in the tank. Make sure the float isn't touching the side of the tank and stopping it from rising. If it is, gently bend the flat rod to stop it from doing this. At the same time check that the float rod isn't broken and that the float itself isn't leaking. If so, it'll need replacing.
Step 2: Flapper Valve
The location of the flapper valve will depend on your toilet.
If you have a siphon toilet, like this one, the flapper valve is actually found in the siphon. You will need to remove the complete siphon if you need to replace the valve.
This will lift open to let the water through into the toilet bowl. Yank this up to let the water flow out. If it doesn't, the flapper valve will need to be replaced. If you have a flapper toilet, it can be found right at the bottom of the tank.
Step 3: Handle
Is it too loose or too tight? Try flushing and adjust the mounting nut if needed. It may also have become stiff with limescale so give it a clean. If it's completely broken you can change it. See "How to replace a broken flush handle" for more help.
Step 4: Chain
This runs from the lift arm down to the bottom of the tank where it lifts the rubbery flapper valve. Check that this is working properly - otherwise this will need to be replaced. It should have about 1/2 an inch of slack, so adjust it by hooking the chain onto a different hole on the handle to get the right tension to sufficiently lift the flapper valve.