Question

Why do electrons (and other very small particles) sometimes behave as particles (i.e. when we are not looking at them) where as other times they behave as waves?

Answer 1

Electrons behave like the same thing all the time. That thing is a solution to a partial differential equation, called the Schrodinger equation. The solution to the Schrodinger equation is called the wave function of the electron.

Although the behavior is always in accord with the same equation, in different circumstances the qualitative aspects of the wave function may be different.

Typical wave functions include gaussians and sinusoids (perhaps inside some sort of wave packet). These functions can either be very narrow or very broad.

When the wave function is broadly-spread compared to the thing it's interacting with, we see significant interference effects. This is frequently called "behaving like a wave".

When the wave function is narrow compared to the thing it's interacting with, we don't see much interference, and this is called "behaving like a particle". Mathematically, though, the wave function (and hence the electron) is behaving in the same way.

There are many applets online that let you play around with wave functions to see how they behave. The first one I found by clicking around is here.