Fig. 1: Axial Force acting on a wood, concrete, and steel column
Axial Stress (aka compressive stress, tensile stress) is a measure of the axial force acting on a beam quantitatively measuring the internal forces acting within in the beam. Compressive stress means the member is in compression (being smashed) vs. tensile stress which means the beam is in tension (being pulled apart).
A couple of important things to note about axial stress are:
 Excluding the selfweight of the beam, the axial stress in a column with no external loads is constant (see Figure 2)
 Except for concrete, tensile forces will normally have greater capacities than compressive forces (think of trying to pull a Popsicle stick apart vs stepping on it and breaking it through compression).
 Axial stress will commonly be used when analyzing columns.
Understanding Axial Stress  
How to Calculate the Axial Stress in a member:
After the axial force of the member is found, Axial Stress (both compressive and tensile stress) are found by taking: f_a = \frac {P_{c/t}}{A}

where:
P_{c/t} = The compressive or tensile force acting on the column (lbs, kips, kgs)
A = the cross sectional area of the column (in^{2}, mm^{2})
Calculating the Allowable Axial Stress for a Member
Fig. 1: Axial Stress in a column is constant (excludes selfweight)
Calculating the axial capacity of a member is all based on the type of material you're working with (and also often dependent on the shape you're using). Concrete has many different ways to design it's columns for compression depending on the length, the type of reinforcing, the amount of flexural stress in the column, etc. Wood uses a bunch of factors to solve for a column's capacity. Since solving for the axial capacity of a beam can be rather complicated, as an example i've detailed the easiest method (wood) below. One reason it's easy imho is because there is an accompanying spreadsheet to do most of the work (always a plus)
For a wooden column:
Most common simplified axial equation: f_a = {F\over A} \le F_a'
F_a' = F_a*C_d*C_M*C_t*C_F*C_i*C_P

All the variables can be found here in greater detail. As you can see even solving for wood is rather complicated.
Example #1
You have a simply supported wooden 8x8 wooden column supporting a ±5 kip load. What is the maximum axial stress (both compressive and tensile) on the column if the column is 12'0" tall?
1) Find the area of the wooden column (remember to use actual sizes)
2) Find the Maximum axial force acting on the column
3) Solve for the tensile/compressive force acting on the beam:
Note: Don't forget to add in the selfweight of the beam.