'Phase' refers to the term encountered while studying waves (in phase, out of phase etc).
When you have a periodic process, phase means how far along it you are. Typically one full cycle corresponds to a phase difference of $2\pi$, because the simplest example of periodicity is moving along a circle, and $2\pi$ is the angle in radians of a full revolution. This is also why you will sometimes see phase measured in degrees.
This is really a very general definition, because whenever you have something that repeats you can think of it as being on a circle, since of course a circle repeats when you go around. If you have a wave that oscillates up and down like a rope, you can think of the y-coordinate as being the vertical coordinate of a particle going around a circle.
A closely related meaning is when you have two copies of a periodic process. In this case, the phase difference (sometimes abbreviated to just phase) measures just that, how much farther along one of the waves (or whatever) is. Two things that are in phase are oscillating together; two things that are $180°$ out of phase are at exactly opposite points in their cycles.
A simple example of two waves being out of phase would be a sine wave and a cosine wave. They are out of phase by pi/2 or 90 deg. Also see this Wiki article https://en.wikipedia.org/wiki/Phase_%28waves%29
The simplest intuitive example is an army in step. When all of them beat down at the same time the phase is zero. If one makes two rows and as the first starts stomping the other lifts the leg, the two rows are in phase at some degrees, depending on the timing difference, from a few degrees to 180, if one maps the stepping to a sine or cosine wave. They are out of phase if every soldier stomps at his own rate (break step).
Soldiers had to break step when crossing the old arch bridges, because if they were in phase and they matched the bridge resonance, the bridge might collapse.