What is frame of reference equations and how to apply?

A frame of reference in physics means where the coordinate system (x-y-z axes) are located as a function of time. From the observers perspective at the origin (frame of reference), he can write equations describing the moving objects position, velocity and acceleration as a function of time. Here's an example for you:

Try to imagine two different observers, one observer is on a moving train at 30mph, the other observer is on the ground watching. The observer on the train throws a ball at 20mph, the length of the train car. As far as the coordinate system where the observer is standing, the ball is moving at 20mph away from him. But as far as the observer on the ground sees, the observer on the train is moving 30 mph away from him, and the ball is moving at 30 + 20 = 50 mph away from him. The equations that each observer would write to describe the moving ball would be different because of the different Reference Frames (Coordinate Axes) where the observers are a function of time.