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$\mathbf{NP}$ is the class of decision problems whose positive instances have polynomial-length certificates verifiable in polynomial time. Equivalently, it is polynomial time on a nondeterministic computation.

$\mathbf{P}$ is the class of decision problems decidable by a deterministic computation in polynomial time.

A polynomial-time many-one reduction from a decision problem $A$ to a decision problem $B$ is a polynomial-time computable function $r$ satisfying $x\in A$ exactly when $r(x)\in B$.

$\mathbf{P}/\mathrm{poly}$ is the class of decision problems decided by polynomial-size circuit families.