What a TLS self signed certificate is at a mechanical level
People routinely talk about self signed TLS certificates. You use them in situations where you just need TLS but don't want to set up an internal Certificate Authority and can't get an official TLS certificate, and many CA root certificates are self signed. But until recently I hadn't thought about what a self signed certificate is, mechanically. So here is my best answer.
To simplify a lot, a TLS certificate is a bundle of attributes wrapped around a public key. All TLS certificates are signed by someone; we call this the issuer. The issuer for a certificate is identified by their X.509 Subject Name, and also at least implicitly by the keypair used to sign the certificate (since only an issuer TLS certificate with the right public key can validate the signature).
So this gives us the answer for what a self signed TLS certificate is. It's a certificate that lists its own Subject Name as the issuer and is signed with its own keypair (using some appropriate key signature algorithm, such as SHA256-RSA for RSA keys). It still has all of the usual TLS certificate attributes, especially 'not before' and 'not after' dates, and in many cases they'll be processed normally.
Self signed certificates are not automatically CA certificates for a little private CA. Among other things, the self-signed certificate can explicitly set an 'I am not a CA' marker in itself. Whether software respects this if someone explicitly tells it to trust the self-signed certificate as a CA root certificate is another matter, but at least you tried.
Self-signed certificates do have a serial number (which should be unique), and a unique cryptographic hash. Browsers that have been told to trust a self-signed certificate are probably using either these or a direct comparison of the entire certificate to determine if you're giving them the same self-signed certificate, instead of following the process used for identifying issuers (of checking the issuer Subject Name and so on). This likely means that if you re-issue a self-signed certificate using the same keypair and Subject Name, browsers may not automatically accept it in place of your first one.
(As far as other software goes, who knows. There are dragons all over those hills, and I suspect that there is at least some code that accepts a matching Subject Name and keypair as good enough.)