Reaction between an acid and acid
In the case of $\ce{HNO3}$ and $\ce{H2SO4}$, the proton from sulphuric acid protonates the $\ce{-OH}$ group of $\ce{HNO3}$ , which gives $\ce{NO2+}$ , $\ce{HSO4-}$ and water. 
This mixture of nitric acid and sulphuric acid is widely used as a nitrating mixture for nitration of aromatic compounds.

In case of other acid mixtures, no reaction happens, but there is an equilibrium. The stronger acids dissociates almost completely , decreasing the dissociation of the weaker one by common ion effect.