In the reactions with acids that we have looked at in the lab the constant characteristic is the transfer of H+ from the acid to some other substance (base, oxide, etc.). Reactions in which electrons are transferred are referred to as reduction-oxidation reactions or simply "redox". Visually these reactions vary considerably. Color changes, precipitates, gas formation, temperature changes--all are possible singly and in combination.
The substance which loses electrons is said to be oxidized. The substance which gains those electrons is said to be reduced.
How do you know when a reaction is redox? If it does not fit the models we have established for precipitation and acid reactions, it is likely to be redox. The only sure way to determine whether a reaction is redox or not is to check for a transfer of electrons.
Oxidation numbers are useful for exactly this kind of task.