What TLS Certificates Establish
A TLS certificate (historically called SSL) is a digital file issued by a trusted Certificate Authority (CA) that binds a cryptographic public key to a specific domain name. It serves as cryptographic proof that the server hosting the site is the authentic owner of the domain.
Certificate Chains and Root Anchors
Browsers do not keep keys for every site. They store root certificates from a small group of trusted CAs. When loading a site, the server presents a certificate chain: its leaf certificate, linked to intermediate certificates, which link back to a trusted root anchor.
Responding to SSL Errors
SSL errors occur when a certificate is expired, misses the request subdomain (e.g. www vs naked), lacks intermediate links in the chain, or is self-signed without local trust configuration.