A certificate warning is your browser declining to proceed because the cryptographic identity of the server could not be verified. The connection reached the server, TLS began, and the browser stopped.
These are worth understanding individually, because they range from trivially benign to the one thing the whole system exists to prevent - And the browser cannot tell you which you are looking at.
ERR_CERT_DATE_INVALID - Expired
The most common by a wide margin. Every certificate has a validity window, and the moment the end date passes browsers refuse the connection. Not a warning to click past by default: A full stop.
For a site, this is a total outage with a mundane cause. The server is running, the application is healthy, the content is served, and no standard browser will complete the handshake. It persists because renewal is often nobody's specific job, and because automated renewal silently failing looks exactly like automated renewal working until the day it does not.
Before blaming the site: If certificates fail on every site, your device clock is wrong. A certificate valid today looks expired to a machine that thinks it is next year, and it looks not-yet-valid to one that thinks it is last year. This is the single most common cause on devices whose battery has died, and on anything freshly reset.
ERR_CERT_COMMON_NAME_INVALID - Wrong hostname
The certificate is valid but was issued for a different name. Visiting example.com and receiving a certificate for www.example.com does it, as does a shared host serving the wrong site's certificate.
Usually a configuration mistake. It is also, though, precisely what an interception attempt looks like, which is why browsers treat it seriously rather than shrugging.
ERR_CERT_AUTHORITY_INVALID - Untrusted issuer
The certificate was signed by something the browser does not trust. Three quite different situations produce it.
A self-signed certificate. Normal on internal tools and development servers, and meaningless on the public web.
A missing intermediate. The server sent its own certificate but not the chain linking it to a trusted root. Frustratingly, this often works in some browsers and not others, because some clients will fetch the missing link and some will not. A site that works in Chrome and fails in Firefox is very often this.
Interception. Corporate networks that inspect TLS install their own root on managed devices; on an unmanaged device the same interception shows up as an untrusted issuer. On a public network this is the warning to take seriously.
ERR_CERT_REVOKED
The certificate was valid and has been deliberately revoked, usually because the private key was compromised. Rarer than the others and never worth bypassing.
ERR_SSL_PROTOCOL_ERROR and handshake failures
Not strictly a certificate problem - The handshake itself failed. Typically no shared cipher suite or protocol version between client and server, which happens when a server is hardened to modern-only and an old client connects, or when an ancient server only offers protocols modern browsers have removed.
It also appears when something is intercepting connections badly, which is why antivirus products with HTTPS scanning are a common cause.
Should you click through?
Browsers deliberately make bypassing awkward, and that design is correct.
Never bypass on a site where you will enter a password, payment details or anything private, on a public or unfamiliar network, or where the warning is a hostname mismatch you cannot explain. The warning exists precisely for the case where somebody is between you and the server.
Reasonable to bypass on your own development server with a self-signed certificate, or on an internal tool you administer and understand.
In between - A well-known site with an obviously expired certificate - The risk is low but the better move is simply to wait. Expired certificates are usually fixed within the hour, because the failure is total and unmistakable and somebody is already on it.
Certificates between machines
Not all certificate errors are between you and the site. If a CDN cannot validate the origin's certificate, the visitor sees a perfectly valid connection to the CDN and an error page from it - Cloudflare's 525 and 526 are exactly this. The TLS problem is one hop further in and there is nothing on the visitor's side to fix.
If it is your site
Automate renewal, then monitor it, because the automation failing silently is the actual risk. Alert at 30 and 14 days remaining rather than relying on the renewal job to complain.
Serve the full chain including intermediates - Testing in one browser is not sufficient, since clients differ in whether they will fetch a missing link. And do not forget internal certificates: An expired certificate on an internal service can take down a deployment pipeline at the worst possible moment, which is how a short outage becomes a long one.