Troubleshooting a running toilet
Since you have a push-button toilet, you don't have what would traditionally be considered a "flapper".  Flush-arm toilets have a hinged rubber stopper that's attached by chain to the flush lever, which opens when you rotate the handle.  

I am not a plumber - this information I've been able to find by using Google and generally understanding how toilets work

Your toilet has a valve in that big cylinder on the right that acts in the same manner, being pulled up in different degrees when you push one of the two buttons.  Underneath the bell-shaped bit at the bottom, I'd expect to find a gasket that keeps water from trickling into the bowl.  Possibly the gasket is worn or bits of debris have gotten stuck under it, allowing some water to escape.

There's also the possibility that the gasket that seals the hole when the valve is closed is similarly worn in a way to allow some water past it when it's closed.  Either way, turn off the water to your toilet and empty the tank (flushing usually works), then take out the right-hand part and see if you can determine where water is getting past the seals.  I'm not overly familiar with this type of toilet, so I don't know if the seals can be serviced by themselves or if you need to buy a whole new flush mechanism.

The other half is your float and water inlet, you shouldn't need to worry about that if you've ruled it out as the culprit.