One of the most fundamental exercises in most riding disciplines is the circle. As a newcomer to horseback riding, you will likely meet the circle early on in your career, as it is beneficial in many ways to both the horse and the rider. As a veteran rider, you are all-too-familiar with the smooth curves and rounding suppleness that results in your horse after a series of various sized circles.
But what really is a circle? What does it look like when completed correctly and what figures can be classified as NOT circles?
Moving straight but not really...
It is true that riding round and round the ring on the fence line (or rail) is one of the easiest things you can be doing on horseback. The second easiest thing to do (not necessarily well) is to ride a straight line from end to end of the ring. Just point the horse's nose and hang on!
What is perhaps less well known is that moving along a straight line is in fact one of the most difficult movements a horse has to learn. Although moving along a straight path is an easily completed "figure", moving straight correctly is rarely achieved. Watch carefully and you will notice the horse's hind end pointing toward the inside of the ring, or the hind footprints not falling over the front footprints.
Enter the circle!
Just like people, horses have a preferred side and tend to want to bear more weight to that side. They are just as uneven as we are. Becoming more ambidextrous is as long of a process for them as it is for us to learn to use both sides of our bodies. And perhaps ironically, one of the most effective ways to develop better straight lines is to ride a circle.
Why should you even bother with a circle?
The primary intention of riding a circle is to help your horse loosen in the muscles and develop suppleness in his movement. It evens out the horse's ability to bear weight in the hind end and stretches both sides of the horse.
If you feel the horse stiffening on a long line, change course and head into a circle.
If you find your horse is distracted or spooking at something outside the arena, the circle is a tried-and-true method to bring his attention back to the (boring) center of the ring.
If you find your horse being uneven in his striding, or leaning in/drifting out, or moving in an otherwise "crooked" manner, then the circle is just right to help him straighten out through his body.
If your horse is a runner and speeds up with increasing tension, put him on a circle and allow him to slow down thanks to the increased weight bearing of the inside hind leg.
What does a circle look like? 
A correctly ridden circle is even and round. I know - that must sound obvious! However, unless you have spent hours on perfecting the circle, you will agree with me that it is easier said than done!
Regardless of where you position the circle in the arena, it should be evenly spaced and round. You must end the circle where you began it, and the diameters should be even - if it is a 20 meter circle, there should be twenty meters from end to end regardless of where you are currently positioned.
The "NOT" Circle
The "NOT circle" isn't quite nearly as useful as the "NOT Canter"!
There are many variations of the not circle - and all of them are not circles! 
A - This circle is one of the most common not circles mainly because of its pseudo-roundness. While you are riding the figure, you are quite sure that you have completed a round figure. That is, until you either look at the footprints in the sand or listen to your instructor's feedback! This circle does not start nor end at the same place and isn't quite evenly round. The horse probably fell in to the middle shortly after the beginning of the circle.
B - This is another common not circle because it is so easy to lose sight of the second half of the circle. Riders often start with good intentions (staying round through the first two quadrants of the circle), and through various inaccuracies - maybe the horse falls in to the middle, or the rider pulls on the inside rein too strongly - the circle ends in an abrupt straight line.
C - Here is another common error - the circle that follows the rail. In the end, you discover that you made a rectangle that basically left one rail and headed straight to another. This figure completely negates the purpose of the circle as the horse doesn't bend through the body. On the other hand, a well-ridden square - OFF the rail - is an extremely beneficial exercise although not at all what we are discussing here.
D - Despite the fact that this final not circle is ever so close to being true, it is not even through the quadrants and therefore ends up becoming more of an oval than a circle. Once again, the horse can avoid bending on the long sides and likely uses the rail as a guideline on where to go.
Parting Thoughts
"The intended effect of working on circles can be achieved only on condition that the correct line of the circle is followed as accurately as possible, but it is difficult to convince riders of the importance of accuracy. Many want quick and easy results and soon lose heart when they discover that riding a correct circle is not as simple a matter as it seems." - Alfred Knophart, Dressage: A Guidebook for the Road to Success (p. 31)
So get out in the ring and be the rider who practices circles to perfection. Work on developing your horse's suppleness and bend, and help him learn to carry more weight on his inside hind leg. Learn the circle aids well and soon enough, "drawing" round, even circles in the sand will become (almost!) second nature!
Finally! The Ultimate Rider-Centered Program!
