the class P
If you had to draw one line on the whole map of computation to separate 'we can really do this' from 'we cannot', complexity theorists draw it at P. P is the class of all decision problems that a deterministic machine can solve in polynomial time. It is the formal home of the everyday word efficient, and almost every algorithm you would call practical, from sorting to shortest paths to search engines, lives inside it.
Precisely, a language L is in P if there is a deterministic Turing machine and a constant k such that the machine decides L (always halts with the correct yes/no) and runs in time O(n^k) on every input of size n. Equivalently, P is the union of TIME(n^k) over all fixed k. The key feature is that the exponent k is a fixed constant that does not grow with the input: n^2, n^3, n^100 are all polynomial, but 2^n is not. P has two properties that make it the right notion. It is robust: every reasonable deterministic model of computation (multitape machines, RAM machines, ordinary programming languages) simulates the others with only polynomial overhead, so membership in P does not depend on which one you pick. And it is closed under composition: feeding the polynomial-time output of one P algorithm into another stays in P, which lets us build big efficient procedures out of small ones.
Two honesty notes anchor P. First, P is defined for decision problems (yes/no questions phrased as languages); a search or optimisation task is studied through its decision version. Second, P is an idealisation of 'feasible', not a perfect synonym for it. An algorithm of time n^100, or one with a hidden constant of 10^40, is technically in P yet useless, and worst-case polynomial time can hide good or bad typical behaviour. So treat P as the standard, well-behaved stand-in for tractable, understood rather than worshipped.
REACHABILITY (given a graph and two vertices s, t, is there a path from s to t?) is in P: breadth-first search answers it in O(n + m) time, polynomial in the graph's size. So is PRIMES (is N prime?), shown by the AKS algorithm. By contrast, no polynomial-time algorithm is known for SAT, so SAT is not known to be in P.
Reachability, sorting, shortest paths, and PRIMES are in P; SAT is not known to be.
P is defined for decision problems on a deterministic machine, and the exponent k must be a fixed constant; 2^n is not polynomial. P is the standard model-independent stand-in for 'efficient', but an n^100 algorithm is still in P and still useless.