Polynomial time
Also known as: Poly time
Polynomial time means running time bounded by a polynomial in the input length: $O(n^k)$ for some constant $k$.
P is exactly the decision problems with deterministic poly-time algorithms. NP uses poly-time verification of a certificate. Poly-time reductions are the standard way to compare hardness inside these classes.
In practice “poly time” is the theory cut between scalable algorithms and combinatorial explosion. A high-degree polynomial can still be slow; exponential is the usual wall for NP-complete search.