a conditional statement
Almost every rule we live by has an 'if...then...' shape: if it rains, then the match is cancelled; if you are under eighteen, then you cannot vote. Mathematics runs on the same shape, and naming it precisely is the first step in reasoning carefully.
A conditional statement has the form 'if p, then q', where p is the hypothesis (the condition) and q is the conclusion (what follows). In symbols, p implies q. The single most important fact about a conditional is when it counts as true: 'if p then q' is false only in one situation — when p is true but q is false. In every other case it is considered true, including, perhaps surprisingly, when p is false (then the statement is vacuously true, because it makes no promise unless p happens). So 'if a figure is a square, then it has four sides' is true, and 'if 2 + 2 = 5, then I am a king' is also true, since its hypothesis never holds.
Conditionals are the basic unit of mathematical claims: nearly every theorem is a conditional, with the hypothesis listing what you may assume and the conclusion stating what you must prove. To show a conditional is false you need just one counterexample — a single case where the hypothesis holds but the conclusion fails. From one conditional you can form three relatives — the converse, the inverse, and the contrapositive — and a conditional whose converse is also true becomes a biconditional, an 'if and only if'.
'If an angle measures 90°, then it is a right angle' has hypothesis p = 'an angle measures 90°' and conclusion q = 'it is a right angle'. It is true. The statement is false only if some angle measures 90° yet is not a right angle — which never happens, so the conditional stands.
If p then q is false only when p is true and q is false; otherwise it holds.
A conditional with a false hypothesis is automatically true (vacuously), and one true example never proves a conditional — but one counterexample disproves it.