Aqua regia is just the name for a mixture of nitric and hydrochloric acid which has the ability to dissolve gold. Aqua fortis was simply an archaic name for nitric acid; acidum salis (literally "salt acid") is likewise an old alchemical name for hydrochloric acid, rather straightforwardly named because it was an acid that was synthesized from salt. (If you ever see hydrochloric acid referred to as "muriatic acid", this is just a somewhat more modernized version of the same name: brine-y/salty acid.)

Oil of vitriol is sulfuric acid. This was also quite a logical name at the time - it's an oily liquid that was made from vitriol (a common term for a class of brightly colored metal sulfate crystals).

So, in the reaction given, when the sulfuric acid is added to saltpeter (potassium nitrate), the sulfate ion swaps places with the nitrate ion and you instead get nitric acid and potassium sulfate.

Source: Have spent far, far too much time researching alchemical nomenclature because I thought it was cool.