What's the difference between dim and bright light?

Brightness is just the number of photons per second hitting your eye - all the other properties of the light are the same.

edit: perceived brightness is the number of 'detected' photons hitting your eye per second!

Different wavelengths of light correspond to different colours. 555nm means light with a wavelength of 555 nano-meters (billions of a meter), this is roughly green light. So all this says is that you eye is most sensitive to green light and so a given number of green photons/second will appear brighter than the same number of red photons. You can see this with laser pointers, for the same power small pointers - green ones look much brighter than red.

The 100 W light bulb dissipates more energy per second (1 watt = 1 joule per second) than the 20 W light bulb, and consequently the light emanating from the 100 W bulb carries more energy than the light emanating from the 20 W bulb.

In the picture of light as an electromagnetic wave, the energy carried by the light is proportional to the square of the wave's amplitude. The technical term for this energy is "Poynting flux". (In fact we usually take the time-average over one period of oscillation as the definition of the energy in the wave.) In this model, the photo-receptors in your eye are oscillators. What is oscillating? Electric charge. Charges are accelerated in response to the electric field of the light: the greater the electric field (or amplitude), the greater the amplitude of the oscillation, and the greater the electric currents in your eye (and the greater the brightness).

In the picture of light as a particle (a photon), each particle carries with it an amount of energy proportional to its frequency: $E=h\nu$, where $h$ is Planck's constant, and $\nu$ is the frequency of light. The energy flux is then the energy per photon multiplied by the flux of photons (# of photons per unit area per second). So the 100 W bulb emits more photons per second than the 20 W bulb. In this model, the photoreceptors in your eye undergo chemical reactions as a result of absorbing photons. The more photons absorbed per second, the brighter the light appears.