First off, a column is a vertical structural element which transmits the weights of the floors above it. But that is not the complete definition. A column, depending on the application, can be subject to tension, compression, shear, bending and torsion. Sometimes multiple types of stresses will be acting at the same time (compression and bending are a very common combination).
For the time being, this section will discuss axial forces acting on a steel member, specifically tensile and compressive forces.
Steel column design has several different checks and criteria, the common ones will be explained further here: