Public key certificate issued "to oneself": the issuer equals subject. The issuer's signature can be verified using the public key contained in itself. This form is typical for root Certification Authorities.
A certificate signed by the private key associated with that certificate. In an X.509 Public Key Infrastructure, all certificates need to be signed. Since root certificates have no third-party signature to establish their authenticity, they are used to sign themselves. In such a case, trust in the certificate must be established by some other means.
Self-signing is one way for the owner of a certificate to sign it themselves instead of having a recognised certification authority do so. In a self-signature the certificate is digitally signed by the very same key whose public part appears in the certificate data. In other words, the signature can be verified using the public key contained in the certificate. This is unlikely to be trusted by anyone wishing to use the certificate as proof of ownership of the corresponding public key. However, a signature by the owner is still useful, especially when the owner is a certification authority which must be trusted for independent reasons, as it restricts the possibilities for malicious or accidental changes to the details contained in the certificate. Compare "Private Certification"
In cryptography and computer security, a self-signed certificate is an identity certificate that is signed by its own creator. That is, the person that created the certificate also signed off on its legitimacy.