An exosphere is the outer portion of an atmosphere, where the material is so sparse that collisions between particles are rare enough to be of little moment. A high frequency of collisions in a gas is part of what makes it "gas-like", with its relation between density, temperature and pressure. I imagine the exosphere particles have orbit-like trajectories, though they undoubtedly are often sufficiently elliptical that they collide with whatever is a level below the exosphere.
What's below the exosphere can be the gas of the atmosphere (e.g., Earth), or the entire atmosphere may be so thin that the exosphere reaches the surface (e.g., Mercury).
The exobase is the border between the exosphere and the normal, gas-like atmosphere below.