In designated verifier signatures, a signer signs a message, byusing her secret key and designated verifier's public key. It can satisfy several security properties, the most common ones being unforgeability (given only randomly chosen public keys of the participants, it is computationally hard to forge signatures) and non-transferability (the designated verifier can, given his secret key and signer's public key, generate a valid signature).
Some more advanced requirements are: private verifiability (to verify a signature, one must know the secret key of the designated verifier), non-delegatability (one can forge signatures only if one knows either of the two secret keys --- this is a stricter requirement than unforgeability!), disavowability (given a valid signature, produced by the designated verifier, signer can prove that it was produced by the designated verifier). Not all DVS schemes satisfy these requirements. (See [LipmaaWangBao2005] for discussion about non-delegatability and disavowability.)
See [JakobssonSakoImpagliazzo1996] for the original definition. A chameleon signature scheme is like a designated verifier signature scheme, except that the designated verifier will be able to forge a signature only after seeing a valid signature on some message. This makes it possible to construct much more efficient chameleon signature schemes. See [KrawczykRabin2000] for the original definition.