A "strong" acid simply means that when added to water, the acid will dissociate completely, therefore a strong acid is strong because it dissociates completely in aqueous solution while a weak acid is weak because it does not dissociate completely in aqueous solution. There are only 7 common inorganic strong acids so I would simply memorize them.
As far as why they are strong acids i.e. why they dissociate completely while other acids do not is a little more complicated and to be honest I am probably not the person to ask. I haven't looked into inorganic acids too extensively, though organic acids you must look at the stability of the conjugate base. A more stable conjugate base means the acid will more easily donate a proton. I do know for a fact that the conjugate base of sulfuric acid exhibits resonance which is a major stabilizing factor. However, based on the structure of sulfurous acid I would expect it to also exhibit resonance, though I could be wrong.
The acid dissociation constant Ka is determined experimentally. Strong acids have a Ka that is >> 1 while weak acids have a Ka that is << 1.
Hopefully I answered your question, but if not I'm sure someone with a little more inorganic chemistry experience will fill in the holes.