I don't know what form active means for a structure. But axial internal force is a force along the line of the member, these forces elongate or shorten the members:

Force ---> ------------ <---Force
This member is in axial compression

Force <--- ------------ ---> Force
This member is in axial tension

A bending force occurs when the applied force occurs perpendicular to the line of the member. This kind of force causes the member to bend.

force
^ --------

Form active structures are structures like domes, cable nets, tensioned fabrics, etc that are sometimes flexible, and either shaped or of material such that under load no bending stress exists, only compression and tension stresses. Rigid domes and arches will have compressive stresses, while fabrics and cable nets will have only tensile forces (flexible cables and fabrics have no ability to resist compressive loading). In your case, it sounds like your professor is teaching you about arches and domes, which are rigid form active structures. These structures, when optimized will have little or no bending stress and only take load in compression.

axial force is force that runs along the length of a member, for example if you have a column, and you put a bunch of load on it, the force of the weight you placed on it will be an axial force.

You should check the "bending force" again. I suspect what you are referring to is bending moment or bending stress. Most engineering terms have specific meanings, and not interchangeable, so it is important to learn the definitions. For bending to be meaningful, it requires a dimensional parameter, such as feet or inches.

If you would like more in depth help, try scanning an image of the lecture notes or give more information, right now your question is pretty vague.

One way to keep track of these things is to follow the units,

Forces will have units like kips, lbs, or newtons

while bending moments have units like kip*ft, kip*in, lbs*ft, or newton*meters.

and stresses will have units like lbs/in^2 (Psi), or Kips/in^2(Ksi)