What is a Phase? What is a Phase? How can we in plain words describe what phase along with in phase and out of phase are, for example while considering a simple pendulum?
 A: The phase can be thought of as the "starting position" of a system like a pendulum - that is, where it is at $t=0$. For instance, if two identical pendulums are started at $t=0$ in the exact same position, they will still be in the same position at some future time $t$, no matter how far ahead we look. We say that these pendulums are in phase, because their motion is perfectly correlated. On the other hand, if you start the pendulums at different times (say that pendulum $1$ is always a half-second ahead of pendulum $2$), then pendulum $1$ will remain ahead of pendulum $2$ forever - the two pendulums have identical equations of motion, but they are out of phase because they will never be in the same place at the same time. For non-identical pendulums (with different periods), the terms "in phase" and "out of phase" simply have no meaning because they will move in and out of phase over time.
Mathematically speaking, the equation of motion for a system like a pendulum (a Simple Harmonic Oscillator) is proportional to $f(t)=\cos(\omega t+\phi)$, where $\phi$ is the phase. Because the cosine function is periodic with a period of $2\pi$, if we have two identical SHOs with respective phases $\phi_1$ and $\phi_2$, they will be in phase if and only if $\phi_1-\phi_2$ is a multiple of $2\pi$ (prove this for yourself!). I hope you find this explanation helpful!
