The three conic sections studied in high school have meanings dating back to the early Greeks, and the present names were probably created by Apollonius around 200 BC, perhaps on the suggestion of Archimedes.
The word parabola literally translates "to throw beside". The Greek root bole, which is the ancestor of ball, bowl, bubble, and many such words today was closer to throw or place in the Greek usage. Symbol is another word with the ball root. The literal meaning is "to throw together", a matching of an image and an idea.
Para is a root for beside, as we see it in parallel. Parallel refers to two lines that are "beside" each other. Think of a parabola as the shape obtained when the cutting plane is "thrown" alongside the slope of the cone. A closely related word is parable, a story that is not true, but true-like, or "thrown beside" the truth. The parathyroids are so named because they are beside the thyroids.
The hyperbola shares the common bola root, but this time the throw is "more than enough" or beyond the slope of the cone, creating a hyperbola. Hyper as more than enough or an overabundance appears in the names of objects with more than three dimensions, and the fourth dimension equivalent of a cube is called a hypercube. Hypertension means too tense. The opposite of hyper is hypo, as in hypodermic, which translates to "under the skin", and of course, Hypotenuse.
The English term, hyperbole derives from the same root words as hyperbola, and is used for an obvious exaggeration or extension of the truth.
The ellipse is a shape formed when the cutting plane of a cone falls short of the slope of the cone, and the meaning, as you expected, is to fall short. The use of three dots or asterisks to mark where part of a statement has been left out is called an ellipsis for the same reason.
The interpretation of falling short of the slope of the cone is NOT the one used by Apollonius in his writing. His reference was to the relationship of the latus rectum. The use of the one above, in the manner of Eutocius, seemed more appropriate for my students.