The fact that there exist oracles $A,B$ such that $P^A = NP^A$, $P^B \neq NP^B$ (a theorem of Baker, Gill, and Soloway) implies that the $P=NP$ question cannot be resolved using "standard" methods (because they would relativize to any oracle).

Edit: Most of the methods used to prove inequalities of complexity classes (e.g. diagonalization in the proof of the space and time hierarchy theorems) work even if the model of computation is changed to allow for one bit of computation to become "free" (i.e. if the Turing machine can access an oracle). The fact that such standard "relativizing" methods cannot suffice (by this theorem) to prove either $P=NP$ or its negation is taken as evidence that $P=NP$ is a nontrivial, serious question. The same is true for $NP$ versus $coNP$. Incidentally, it is true that $P^A \neq NP^A$ relative to a random oracle (where "random oracle" means that membership of any string in $A$ is decided randomly) with probability 1,* though this does not mean that $P \neq NP$.

An example of a nonrelativizing method is the arithmetization of boolean formula (which converts a boolean formula to a polynomial which is nonzero at an assignment of the variables to zeros and ones iff the formula is satisfiable with those as the corresponding inputs), used in the proof of Shamir's theorem that $IP=PSPACE$, a fact not true relative to arbitrary oracles.

*The probability has to be zero or one by Kolmogorov's zero-one law, incidentally.

There is a very interesting discussion of the Baker-Gill-Soloway theorem on Terence Tao's blog, here.