Are the distances between two celestial bodies the distance of their centers? For example, the average Sun-Earth distance is 1 AU. Is this


the distance between the center of the Sun and the center of the Earth
the distance bewteen the surface of the Sun and the surface of the Earth,
or something else?
Distance is an ambiguous term that can mean both centre-centre and surface-surface depending on the context it is used.


In cases where the size of the objects is trivial compared to the distance, 'distance' is almost exclusively used for the "distance between centres" of the two objects. This is the case for Sun-Earth, Earth-Moon and almost any other pair of significant solar system object. The alternative meaning must be specified if intended, like: "the distance from the surface of the Sun to Earth"
It may however also mean the "distance between surfaces", especially when the two objects are very close to each other. This is often the case with binary asteroids.


For  an orbital spacecraft, the word 'distance' is mostly not used when referring to the central object. The terms 'altitude' or 'orbital radius' are preferred. Altitude is the distance to the surface (or rather, to a theoretical sealevel. It may also refer to the actual surface though, so be careful). Orbital radius is the distance to the centre of the Earth.

As a side note "average distance" in an orbit is referring to the semi-major axis as that is the number used for calculating the orbital period and different velocities.