What are frames of reference?

A frame of reference is just a coordinate system for observation.

For simplicity, lets consider a 2-dimensional square in Cartesian coordinates (just the usual x and y axes). We'll say the square has a side length of 1. Now picture the square facing you, just floating. Now we are going to apply the x and y axes. The x axis will be horizontal and the y axis will be vertical. The point where they meet is called the origin, (0,0).

How you place the origin and the two axes is what is considered your frame of reference. If we put the origin at the center of the square with our x axis pointing horizontally through one of the side of the square and our y axis pointing vertically through the other two sides of the square, then we the coordinates of the four corners would be (1/2,1/2), (-1/2,1/2), (1/2,-1/2), and (-1/2,-1/2). We could have put our origin at the bottom left corner of the square with the x axis going through the bottom edge and the y axis going through the left edge. Then our four corners would be (0,0),(1,0),(0,1),(1,1). Both of these are equally valid frames of reference.

To summarize all of this, your frame of reference is where you define the origin, the (0,0) point, and the orientation of your axes. We could spin our axes so that its going through the corners of the square. We could shift it far away from the square; it doesn't matter what you do, the length of the square will remain 1.

Enough of the 2-D case. Picture a person, person A, standing on the earth, looking into the sky. Person A will see clouds moving flowing by, and after some time, that the stars are spinning. Now zoom out into space, perhaps by the moon, and look at person A. Now you'll see person A spinning around as the earth spins around, but if you look at the stars, you would barely see any change. Definitely not spinning like person A saw.

So what explains this? The change in reference frame between you and person A. Person A's perspective is known as a rotating frame of reference. They are located on a spinning body (the earth), but wouldn't even know it (unless they were looking for it). The whole world used to think the stars were rotating, until someone dared to think outside their reference frame.

So this is the point where math takes over. Instead of Cartesian coordinates (x,y,z) you can get into cylindrical (r, theta, z) or spherical (row, theta, phi) or even crazier ones. It takes several years and a lot of studying of physics and math to handle the majority of coordinate systems and transformations between them. Definitely worth it though!