Strong acids are acids for which the equilibrium constant for dissociations is so large that we assume the molecules "fall apart" or dissociate 100% in water.
For example, HCl is a strong acid. So for the equilibrium
\[\rm{HCl(aq) + H_2O(l) \rightleftharpoons H_3O^+(aq) + Cl^-(aq)}\]
we assume the equilibrium constant is so large (it is more than 106) that the reaction will essentially go to completion. That is when we put HCl in water, it will completely dissociate and form H3O+ and Cl-. This is what we mean by strong. We call it an acid because it is donating a proton (H+) to water.
You need to memorize the list of strong acids. Both their names and formulas
Hydrochloric acid - HCl
Hydrobromic acid - HBr
HydroIodic acid - HI
Perchloric acid - HClO4
Chloric acid - HClO3
Sulfuric acid - H2SO4 (note only the first proton is strong)
Nitric acid - HNO3