Mercer Theorem

If $K(u,v)$ is a continuous, symmetric function (which means $K(u,v)=K(v,u)$) and positive semidefinite, then there exist a mapping $\Phi (x)$ such that:

$$K(u,v)=⟨\Phi (u),\Phi (v)⟩$$

and $K$ is a valid kernel.

Positive Semidefinite (PSD)

The condition is:

$$\int \int K(u,v)g(u)g(v)dudv>0$$

Basically it means that the two functions will never produce a negative similarity energy. Energy here refers to something like squared length.

The reason why we need PSD is because without PSD, we could have some squared lengths ${\Vert \begin{array}{c}\Phi (x)\end{array}\Vert }^2=K(x,x)$ that could be negative. Which is wrong.

You can read more in Functional Analysis textbooks.