Hi araan.
Allowable stress is a term used to indicate the maximum amount of stress a material may be subjected to given a predetermined factor of safety. In mechanical engineering, we use allowable stress in piping and pressure vessel analysis. I believe civil engineering uses this concept for other purposes as well, including reinforced concrete structures.
Regardless of its use, the calculation of stress can be done using any method suitable, such as conventional hand calculations, using Roark, ASME code calculations, FEA, etc... The stress obtained from those calculations must then be compared to the maximum stress the material is allowed to be exposed to (ie: the allowable stress). For piping codes for example, the stress determined by calculation may not exceed 2/3 of yield strength or about 1/4 of ultimate tensile strength, depending on the code.