What defines a planet's orbit?
planets are revolving in an elliptical orbit depending upon their mass and velocity they got at time of formation.
Planets orbit in elliptical orbit.. Circle is a very special case of ellipse with zero eccentricity. Planets follow Kepler's first law..
An orbit is defined by the orbiting body's distance from the parent and its velocity.
When a body is orbiting in a gravitational field, the path it follows is determined by its position and velocity. The initial velocity has to be tangental for a circular orbit which is a very unlikely event.
A planet in a stable orbit has an elliptical orbit with the parent body being at one of the foci. Kepler proved this by geometry, Newton proved it using calculus, Einstein refined it as being a geodesic in curved space time. A geodesic is the 4 dimensional equivalent of a straight line.
A circle is a special case of an ellipse, where the two foci are at the same point. Circular orbits are uncommon in nature as the conditions have to be just right.
In our solar system only Venus and Neptune have orbits which are very near circular. Comets have very elliptical orbits.