We see continuous spectra (ie white light) from thermal sources (ie blackbody radiation). The emitted photons get their energy from the kinetic energy of the atoms in the source. Basically the frequency of an emitted photon depends on the velocity of the particle that emitted it. Since a particle can have any velocity (below light speed of course), and there is no quantization, the emitted photons can have any frequency. Add up a whole ton of particles radiating this way and you'll get a continuous spectrum.
Hydrogen on the other hand radiates in a specific way. Hydrogen is a bound state of an electron and a proton and this system is governed by quantum mechanics (it's a good thing too... if it wasn't, all atomic matter in the universe would decay in about 10^-20 seconds or so). The atom radiates when an incoming photon knocks the electron into a higher energy level. This photon can have any energy and the atom will re-radiate this energy, provided that the photon isn't energetic enough to completely ionize the atom (blows the electron right off the proton). The electron then falls back to the lowest energy state in the atom, radiating energy as it goes. However, because of quantum mechanics, there are only certain discrete levels it can occupy. For example imagine you are climbing a ladder. You can be 1 rung, 2 rungs, 3 rungs from the ground, but because there is no 2.223rd rung, you can't be that far from the ground. The energy levels become pretty complex once you take into account all of the factors, but basically they follow a law like:
E(n) = (constants like c, hbar, 2pi, electron charge squared) x (1/nf^2 - 1/ni^2) where ni and nf are integers. You can see that completely ionizing a hydrogen atom (ni = 1, nf=infinity) requires 13.6 eV of energy. You can find the energy difference between any two levels using this formula. Of course it gets a little more complex, as each of these levels undergoes some splitting, but that's probably not required in your explanation.
It is because of this quantum mechanical nature of small things that hydrogen emits radiation only in discrete steps. This is where the good thing comes in. If the electron could radiate away it's energy continuously, it would be able to fall into the proton and atomic matter as we know it would not be possible.
edit: better analogy