(Mechanics, Theory of Relativity, etc)
A frame of reference is simply a way to assign coordinates to physical objects. In the 3D space we live in, four coordinates are required to completly specify an event: three space coordinates and the time. In principle, any three unrelated quantites will do for the space position, but normal Cartesian coordinates are often used.
An example of a frame of reference would be: position is described by the distance North and South of the Eiffel tower, and the height above or below its tip. Time is described by years A.D.
When you only want to specify the place and time of some event, any frame of reference will do, although some will be more convenient than other. However, when it comes to actually trying to predict how an object will behave, i.e. formulating laws of motion, some frames are much worse than others. For example, if we choose to measure height from a bouncy ball, we have to explain why the everything is bouncing up and down (except the ball, which is at rest). And if we choose the above "Eiffel tower frame", we have to explain why the universe is rotating (one revolution per 24 hours). There is a particular type of frame, which gives the simplest equations of motion: the inertial frame of reference.
Frames of reference are used in the more theoretical formulations of Newtonian mechanics, but their properties there are so intuitive that you can keep them implicit in practice. However, the essential contribution of the theory of relativity is that translating coordinates between frames moving with high relative velocity gives some surprising results. When relativity became a standard theory, it also created interest in frames of reference, and they are now discussed in books about non-relativistic physics too.