Geostationary communications satellites do have a basically circular orbit, as you said, but many other birds do not.
Russia has a bunch of stuff in orbits that "linger" very high over Russian territory for much of their orbital period then shoot right around close to the back side of the planet quickly and linger again, these are called Molniya orbits.
The US has a bunch of secret (presumably spy) satellites that fly less obvious orbits like this too, for presumably similar reasons.
† Edited: This post originally said "very" different, but in fact the difference would be modest since our orbit just isn't that elliptical. I wasn't able to find out how modest, but certainly if you think summer in New Zealand (in December) is pretty similar to summer in England (in June) then it's reasonable to say at least that similar.
Geosync is the altitude at which a circular orbit takes 24 hours - meaning it takes as long to complete one orbit as it takes the earth to complete one revolution.
But there’s an orbital velocity at EVERY altitude, and therefore there’s an orbital period at every altitude too.
If you are at a particular altitude and you are going at a different speed than the one needed for a circular orbit, you are either going too fast (in which case you are in an elliptical orbit and you will gradually increase altitude and lose speed until you reach the top of that ellipse) or you are going too slow to stay in a circular orbit (in which case you will follow an ellipse down to a lower altitude and gain speed).
In that ‘too slow’ case, if it’s much too slow then the ellipse you follow will take you low enough to hit the surface or atmosphere of whatever you’re orbiting (ie you will crash into it)