Contradiction, counterexample, when and how to use each form of proof and disproof?

CounterExample can be used to disprove a universal statement saying something like "For all integers....if [something] then [something]" a way to counterexample is to simply find one situation or one set of numbers where the statement is false.

Can someone explain to me how to use contradiction?

Also, what would you use in situation where the statement is completely false? counter example wont really work for that.... I was wondering if something like this will ever pop up during a typical discrete mathematics course?