A dramatic example of non-inertial frames can been seen in accelerating rockets.

When a rocket accelerates, an occupant feels "gravity".

 

Consider a rocket with a small "window" in one side and a major league pitcher with amazing accuracy.
 

An outside observer sees a horizontally thrown ball travel in a straight line.  Since the ship is moving upward while the ball travels horizontally, the ball strikes the wall somewhat below a point opposite the window.

To an inside observer, the path of the ball bends as if in a gravitational field.

From this thought experiment we realize that we cannot tell the difference between an accelerating frame and one that is at rest and has a gravitational pull.  This is the "equivalence principle" and is the corrnerstone to Einstein's theory.
 
 
 

Now replace the ball with a light beam.  Again, to the observer in the rocket it will apear to have it's path "bent."

This tells us that gravity "bends" light beams.  Stars, including our sun are massive enough to bend light from distant stars.
 

But, wait a minute... doesn't light travel in a stright line?  How can this be?
 
 
 

Perhaps space is somehow "curved"  If this is the case then light still travels in a stright line, but now the idea of "striaght" has to change since our surface is no longer flat.  Think about how we define parallel lines on a flat surface and one a sphere. 
 
 
 
 

Einstein's theory of special relativity talks about how space and time are connected.  This theory deals with objects and reference frames moving at constant speeds.  His theory of general realtivity deals with accelerating objects (which includes gravity).  The mathematics of special relativity isn't too bad, but the mathematics of general realtivity is a matter for a graduate level course.  Both theories are well establish experimentally.