A great circle results from the intersection of a plane which cuts a sphere right through its center, much like when you cut an orange in two equal parts. In other words, it's the greatest possible circle on a spherical surface, hence its name. All other circles that don't pass in the center of a sphere have a smaller circumference and belong to the small circle family. Setting our minds into a greater scale, the equator is a great circle whereas all other latitude lines fall into the small circle category.
The euclidean space seems so familiar to us that we blindly accept the shortest distance between two points as being a straight line, this is not true though for other spaces. Actually in a spherical surface the shortest path between two points is far from being straight. In order to find the smallest possible distance connecting two points you need to find the particular great circle which not only cuts the sphere in two halves going through its center but also passes by these two points. Thus, the equivalent of a straight line between point A and point B in euclidean space is an arc along a particular great circle in spherical space.
An interesting practical application of great circles can be seen in aviation routes for long haul flights provided you ignore favorable winds. Often this routes are plotted on maps as curves rather than straight lines going against our intuition of shortest path, it turns out to be a consequence of projecting a spherical space into a plane. The familiar curved paths on the map are in fact great circle paths going through source and destination airports, using straight line routes would span out unnecessarily longer distances. This may seem hard to grasp at first since we're mentally stuck into euclidean space, having said that there's an entertaining experiment you can try at home.
Get an Earth globe and a piece of sewing line, then pick up two distant world cities laying in the same hemisphere like for instance Lisbon, Portugal and Los Angeles, United States. Next place one end of the sewing line at one of the cities and stretch it to the other city making sure it's really tight and can't be further stretched. As you can see for yourself the resulting route surprisingly starts out by climbing a few degrees in the latitude scale at both ends and goes as far high as Newfoundland in Canada, something you would never even dream while trying this in a flat map. Try out different source and destination cites noticing the dramatic effects as you increase the distance and move away from the equator towards the poles, at some point the great circles reach the North or South pole. On the other hand the closer to the equator the similar it gets to a straight line in the map since the equator is itself a great circle.