Bohr's second postulate What was Niels Bohr's reasoning behind his second postulate (about integer angular momentum)? How did he come up with it without any knowledge of matter waves?
I only have a high school level education so go easy on me.
 A: First, let's just write out Bohr's second postulate so we are on the same page:

Bohr's second postulate: The angular momentum of an electron in orbit around the nucleus of an atom is an integral multiple of $\hbar$, where $\hbar=h/(2\pi)$ and $h=6.6...\times10^{-34}{\rm m^2\ kg\ s^{-1}}$ is Planck's constant.

I won't actually try to answer the question "what was Bohr thinking when he came up with this?" For that, you are probably better off asking at the history of science stack exchange. However, I can offer some sense of why, from a physics point of view, such a postulate is useful for explaining the motion of electrons in the atom.
Through a fascinating series of experiments in the early 1900s, physicists in the mid 1910s were led a picture of the atom in which a negatively charged, light, electron orbited a heavy, positively charged nucleus. The simplest version of this picture would be a "planetary model," in which the electron is like a little Earth orbiting the nucleus, which is like the Sun.
However, this picture fails miserably, when we take into account the theory of electrodynamics. Maxwell (and others) showed that an accelerating charge gives off energy in the form of electromagnetic radiation, or light. As the charge emits energy, its orbit will become smaller and smaller, and eventually the electron will plunge into the proton. It is a fun calculation to estimate how long this process takes; suffice to say it is much less than one second, and this model of the atom is hopelessly, hilariously bad since it would imply that all atoms (and therefore all matter) should have collapsed long ago.
Therefore what a theoretical physicist in the mid-1910s needed was a way for electrons to orbit the atom (which was the picture that explained the available experiments), without losing energy due to electromagnetic radiation. A wild, speculative, crazy idea (at least for the time) would be to say that the electrons can't emit an arbitrary amount of radiation, because they can only exist in certain, well-defined orbits. In particular, if there was a smallest allowed orbit, this would allow electrons to orbit the nucleus, while protecting them from orbital decay.
Bohr's second postulate is then a way of implementing this idea. By requiring the angular momentum to be quantized, Bohr is saying that only certain orbits of the electron are allowed to take place. In particular, there is an orbit with a smallest amount of angular momentum (with angular momentum $\hbar$), and the electron cannot emit radiation to move to a smaller orbit than this, and cannot crash with the proton. This provides stability for the electron.
You might ask, "why angular momentum"? Part of the reason is the correspondence principle. Bohr reasoned that whatever rules were invented to handle the motion of electrons in the atom (in a very "quantum" regime), must reduce to classical physics in the limit that $\hbar \rightarrow 0$ (when quantum effects are small). By quantizing angular momentum in units of $\hbar$, Bohr satisfied this principle, since angular momentum is a continuous variable in classical physics. Or, stated differently, the relative difference between $n \hbar$ and $(n+1) \hbar$ becomes very small as $n$ becomes large, so angular momentum is continuous to a very good approximation for systems with a macroscopic amount of angular momentum. Angular momentum is also a good choice of quantity to apply Bohr's quantum rule to because it is conserved classically (in fancier terms it is an adiabatic invariant); angular momentum is not just some random function of the position and momentum, but is a property of the system that has a meaning that persists in time.
It should be noted Bohr's (and Sommerfeld's) set of ad hoc rules get the right answer, and were an important stepping stone in the development of quantum mechanics, but they did not have a rigorous explanation and the rules only worked in special situations. Ultimately this picture is justified in quantum mechanics, which superseded the rules, and from which their rules can be derived in the so-called semi-classical limit.
