What is a parabola?

A parabola is the locus of points that is equidistant from both a certain point (called the focus) and a certain line (called the directrix).
It is the shape of the continuous function
##y(x) = ax^2 + bx + c##
or with reference to more typical physical notation
##y(t) = y_0 + v_{0y} t + frac{1}{2}a_y t^2##
(for the case of subject to a single force acting vertically).
A parabola is also one of the allowed shapes of orbit for motion of a particle subject to an inverse square force (such as gravity or the electrostatic force). The parabola is the orbit corresponding to zero total (with potential energy defined to be zero at infinity as is conventional for the study of orbits).
A parabola is also a conic-section. (The others are ellispses (including circles hyperbolas points lines and intersecting lines).
(see http://en.wikipedia.org/wiki/Conic_section for more information on conic sections). The parabola is the conic section that is obtained by slicing the cone parallel to the edge of the cone.