Intuitively, NP is the set of all decision problems for which the 'yes'-answers have simple proofs of the fact that the answer is indeed 'yes'. More precisely, these proofs have to be verifiable in polynomial time by a deterministic Turing machine. In an equivalent formal definition, NP is the set of decision problems solvable in polynomial time by a non-deterministic Turing machine.
The complexity class P is contained in NP, but NP contains many important problems, called NP-complete problems, for which no polynomial-time algorithms are known. The most important open question in complexity theory, the P = NP problem, asks whether such algorithms actually exist for NP-complete problems. It is widely believed that this is not the case.
The complexity class NP can be defined in terms of NTIME as follows:
Many natural computer science problems are covered by the class NP. In particular, the decision versions of many interesting search problems and optimization problems are contained in NP.
As the number of integers that we feed into the algorithm becomes larger, the number of subsets grows exponentially, and in fact the subset sum problem is NP-complete. However, notice that, if we are given a particular subset (often called a certificate), we can easily check or verify whether the subset sum is zero, by just summing up the integers of the subset. So if the sum is indeed zero, that particular subset is the proof or witness for the fact that the answer is 'yes'. An algorithm that verifies whether a given subset has sum zero is called verifier. A problem is said to be in NP iff there exists a verifier for the problem that executes in polynomial time. In case of the subset sum problem, the verifier needs only polynomial time, for which reason the subset sum problem is in NP.
Note that the verifier-based definition of NP does not require an easy-to-verify certificate for the 'no'-answers. The class of problems with such certificates for the 'no'-answers is called co-NP. In fact, it is an open question whether all problems in NP also have certificates for the 'no'-answers and thus are in co-NP.
Because of the many important problems in this class, there have been extensive efforts to find polynomial-time algorithms for problems in NP. However, there remain a large number of problems in NP that defy such attempts, seeming to require superpolynomial time. Whether these problems really aren't decidable in polynomial time is one of the greatest open questions in computer science (see P=NP problem for an in-depth discussion).
An important notion in this context is the set of NP-complete decision problems, which is a subset of NP and might be informally described as the "hardest" problems in NP. If there is a polynomial-time algorithm for even one of them, then there is a polynomial-time algorithm for all the problems in NP. Because of this, and because dedicated research has failed to find a polynomial algorithm for any NP-complete problem, once a problem has been proven to be NP-complete this is widely regarded as a sign that a polynomial algorithm for this problem is unlikely to exist.
The two definitions of NP as the class of problems solvable by a nondeterministic Turing machine (TM) in polynomial time and the class of problems verifiable by a deterministic Turing machine in polynomial time are equivalent. The proof is described by many textbooks, for example Sipser's Introduction to the Theory of Computation, section 7.3.
To show this, first suppose we have a deterministic verifier. A nondeterministic machine can simply nondeterministically run the verifier on all possible proof strings (this requires only polynomially-many steps because it can nondeterministically choose the next character in the proof string in each step, and the length of the proof string must be polynomially bounded). If any proof is valid, some path will accept; if no proof is valid, the string is not in the language and it will reject.
Conversely, suppose we have a nondeterministic TM called A accepting a given language L. At each of its polynomially-many steps, the machine's computation tree branches in at most a constant number of directions. There must be at least one accepting path, and the string describing this path is the proof supplied to the verifier. The verifier can then deterministically simulate A, following only the accepting path, and verifying that its accepts at the end. If A rejects the input, there is no accepting path, and the verifier will never accept.
NP contains all problems in P, since one can verify any instance of the problem by simply ignoring the proof and solving it. NP is contained in PSPACE - to show this, it suffices to construct a PSPACE machine that loops over all proof strings and feeds each one to a polynomial-time verifier. Since a polynomial-time machine can only read polynomially-many bits, it cannot use more than polynomial space, nor can it read a proof string occupying more than polynomial space (so we don't have to consider proofs longer than this). NP is also contained in EXPTIME, since the same algorithm operates in exponential time.
The complement of NP, co-NP, contains those problems which have a simple proof for no instances, sometimes called counterexamples. For example, primality testing trivially lies in co-NP, since one can refute the primality of an integer by merely supplying a nontrivial factor. NP and co-NP together form the first level in the polynomial hierarchy, higher only than P.
NP is defined using only deterministic machines. If we permit the verifier to be probabilistic (specifically, a BPP machine), we get the class MA solvable using a Arthur-Merlin protocol with no communication from Merlin to Arthur.
There is also a simple logical characterization of NP: it contains precisely those languages expressible in second-order logic restricted to exclude universal quantification over relations, functions, and subsets.
NP can be seen as a very simple type of interactive proof system, where the prover comes up with the proof certificate and the verifier is a deterministic polynomial-time machine that checks it. It is complete because the right proof string will make it accept if there is one, and it is sound because the verifier cannot accept if there is no acceptable proof string.
A major result of complexity theory is that NP can be characterized as the problems solvable by probabilistically checkable proofs where the verifier uses O(log n) random bits and examines only a constant number of bits of the proof string (the class PCP(log n, 1)). More informally, this means that the NP verifier described above can be replaced with one that just "spot-checks" a few places in the proof string, and using a limited number of coin flips can determine the correct answer with high probability. This allows several results about the hardness of approximation algorithms to be proven.
The decision version of the traveling salesman problem is in NP. Given an input matrix of distances between N cities, the problem is to determine if there is a route visiting all cities with total distance less than k. A nondeterministic Turing machine can find such a route as follows:
One can think of each guess as "forking" a new copy of the Turing machine to follow each of the possible paths forward, and if at least one machine finds a route of distance less than k, that machine accepts the input. (Equivalently, this can be thought of as a single Turing machine that always guesses correctly)
Binary search on the range of possible distances can convert the decision version of Traveling Salesman to the optimization version, by calling the decision version repeatedly (a polynomial number of times).