Here is the relevant quote from RFC 5280RFC 5280.
The signatureValue field contains a digital signature computed upon the ASN.1 DER encoded tbsCertificate. The ASN.1 DER encoded tbsCertificate is used as the input to the signature function. This signature value is encoded as a BIT STRING and included in the signature field. The details of this process are specified for each of the algorithms listed in [RFC3279], [RFC4055], and [RFC4491].
By generating this signature, a CA certifies the validity of the information in the tbsCertificate field. In particular, the CA certifies the binding between the public key material and the subject of the certificate.
Basically, the certificate signature is a CA signed value of the information encoded in the certificate including things like the subject, the issuer and the public key of the certificate. The process of validating the signatures is described in various RFCs as mentioned.
The certificate fingerprint is simply a sha1 and sha256 hash value computed on the entire certificate.
