Val'reth is a Vulcan term given to those members of their kind that possess the katra of another Vulcan in their mind against their will. One known person who suffered this 'state' was T'Prynn, whose would-be mate Sten forced his katra into her mind when she killed him, the katra refusing to leave until T'Prynn had submitted. She was finally 'cured' of this condition via a complex ancient ritual known as the Dashaya-Ni'Var, performed by the Healer Sobon. (VAN novels: Harbinger, Open Secrets)
Given the context of Spock's description, it is unclear whether the term "val-reth" refers simply to the fact that T'Prynn did not volunteer to host the katra- in which case Jonathan Archer and Leonard McCoy could also be described as val-reths as neither of them volunteered to host the katras of Surak or Spock-, or simply refers to the fact that Sten's katra will not leave.