A body in orbit follows an elliptical path with the mass it is orbiting at one of that ellipse's focal points. Eccentricity (abbreviated as e) defines the shape of that elliptical path, and specifically how far it is from a simple circle. An value of zero describes an orbit with no eccentricity at all, or a perfect circle. It is usual for orbital eccentricities to be higher than zero, and as values for eccentricity increase, they describe orbital paths that trace out longer and narrower ellipses.
No known body has an orbital eccentricity of exactly zero, but several planets and moons in the Solar System have orbits that approach that value. Earth, for example, has a low orbital eccentricity of 0.0167, meaning that its orbit around the Sun is nearly circular. The lowest known eccentrity for any body in the Solar System is that of the moon Triton, a value of 0.000016, meaning that it orbits its parent planet Neptune in a very nearly perfect circle.
As eccentricity increases, orbital paths become more distinctly elliptical. Following an elliptical orbit means that a body is sometimes closer to, and sometimes farther from, the object it is orbiting, and at higher eccentricities this difference becomes more and more pronounced. The general terms for the closest and most distant points on an orbit are periapsis and apoapsis respectively, though these generic terms are commonly replaced by terms referring to specific bodies. Objects orbiting the Sun, for example, have perihelion and aphelion points in their orbits, while satellites of Earth have a perigee and an apogee.
For objects with high orbital eccentricities there can be very significant differences between their periapsis and apoapsis. Some extreme examples can be found in the outer Solar System: for example, the probable dwarf planet Sedna has a high eccentricity of 0.8549, and so its distance from the Sun can vary by more than an order of magnitude, between 76 AU at its closest approach and no less than 936 AU at its aphelion.
Periodic comets can have even greater eccentricities. The orbit of the famous Halley's Comet, for example, has an extreme eccentricity of 0.9671, so that while its perihelion brings it well within Earth's orbit, the outer leg of the comet's path takes it to a distance of some 35 AU from the Sun, far beyond the orbit of Neptune. When the eccentricity value reaches one or above, the orbits it describes are not closed orbits at all, but instead describe open paths (for example, non-periodic comets make a curved path around the Sun and then continue out into space, never to return).
Comets provide an example of how highly eccentric orbits can be created, in this case due to their origins in the outer Solar System and the immense gravity of the Sun pulling them inward. Orbital eccentricity is in fact affected by various forces like this; for example, bodies orbiting a massive object will tend towards more circular orbits. Other bodies will also exert their own gravitational effects, such as the influence of massive Jupiter on the orbits of planets and asteroids in the inner Solar System. In combination this complex interplay of effects mean that orbital eccentricity is rarely fixed, and orbital paths can become more eccentric, or more nearly circular, over long periods of time.