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A tautology is always true regardless of truth assignments; a contradiction is always false. Recognizing these is crucial: if assuming negation of a statement yields a contradiction, the statement must be true.
Compare tautologies (p ∨ ¬p) with contingencies (p ∧ ¬q) that can be either true or false.
Every logical statement lives somewhere on a spectrum between always-true and always-false. From your truth table work, you know that a compound statement can be true under some truth assignments and false under others — these are called contingencies and they're the typical case. But at the extremes are two special cases: tautologies are true under every possible truth assignment, and contradictions are false under every possible truth assignment. These are not just statements that happen to be true or false — they're true or false as a matter of logical structure alone, independent of any facts about the world.
The canonical tautology is p ∨ ¬p: "p or not p." No matter whether p is true or false, exactly one of the disjuncts is true, so the whole statement is always true. It captures the law of excluded middle. The canonical contradiction is p ∧ ¬p: "p and not p." This is always false because no statement can simultaneously be true and false. Notice these two are negations of each other: ¬(p ∧ ¬p) ≡ p ∨ ¬p, and a tautology's negation is always a contradiction.
The proof-theoretic importance of contradictions is enormous. Proof by contradiction works as follows: to prove a statement Q is true, assume ¬Q and derive a contradiction — a statement of the form (R ∧ ¬R). Once you've derived a logical impossibility, the assumption ¬Q must be false, which means Q must be true. The classical example is the irrationality of √2: assume √2 = p/q in lowest terms, derive that both p and q must be even (contradicting "lowest terms"), and conclude the assumption was false. The power of this method depends entirely on understanding that a contradiction is not just a "false thing" but a statement that cannot possibly be true under any assignment — so reaching it proves the starting assumption was impossible.
Tautologies play a complementary role as rewriting tools. In formal logic and computer science, rules of inference are tautologies: modus ponens says that ((p → q) ∧ p) → q is always true. When you apply a proof rule, you're instantiating a tautology. Recognizing a tautology tells you that an argument form is universally valid — it works for any specific statements you plug in. Recognizing a contradiction tells you that a set of assumptions is inconsistent — they can never all be simultaneously true — which is why reaching one during a proof demolishes the premises that led there.