in your example, when you get the fast blink on right-turn signal, are the headlamps on? If the lights are on, then probably the flasher filament is gone in that bulb (it's probably a dual-filament bulb - one for parking lights and another brigher filament for the blink). If the lights are off, then something's not right with the wiring there, because if the rear flashes, the front should too.

It may also be a corroded bulb socket.

A flasher works by using the "draw" of the circuit to heat a bi-metalic strip which is attached to a contactor in the flasher. As the strip heats up, it bends and causes the circuit to break. When the circuit breaks, the power that's causing the strip to heat is removed and the strip cools down. Once it cools sufficiently, the circuit re-connects and the process starts all over again.

When a bulb is burnt out or not receiving electricity, the amp draw of the circuit isn't as high, so the element doesn't get as hot as quickly, therefore the cooldown period is quicker - hence the faster flash.