Firefox, along with all other browsers, are trying to move away from supporting older TLS versions, which means means anything before TLS 1.2. In Firefox, the minimum acceptable TLS version is controlled about the about:config preference security.tls.version.min; in released versions of Firefox this is still '1' (for TLS 1.0), while in non-release versions it's '3' (for TLS 1.2). If you're using a non-release version and you visit some websites, you'll get a 'Secure Connection Failed' interstitial warning that's clear enough if you're a technical person.

The bottom of the warning text says:

This website might not support the TLS 1.2 protocol, which is the minimum version supported by Firefox. Enabling TLS 1.0 and TLS 1.1 might allow this connection to succeed.

TLS 1.0 and TLS 1.1 will be permanently disabled in a future release.

It then offers you a big blue 'Enable TLS 1.0 and 1.1' button. If you pick this, you're not enabling TLS 1.0 and 1.1 on a one-time basis or just for the specific website (the way you are with 'accept this certificate' overrides); you're permanently enabling it in Firefox preferences. Specifically, you're setting the security.tls.version.enable-deprecated preference to 'true' (from the default 'false').

As far as I've been able to see, the state of this '(permanently) enable deprecated TLS versions' setting is not exposed in the Preferences GUI, making its state invisible unless you know the trick (and even know to look). Perhaps when Mozilla raises the normal minimum TLS version in a Firefox release, they will expose something in Preferences (or perhaps they'll change to do something with per-site overrides, as they do for TLS certificates). In the mean time, if you want to find out about websites using older TLS versions through your normal browsing, you'll need to remember to reset this preference every time you need to use that big blue button to get a site to work.

(You might be doing this in Nightly or Beta, although probably you should avoid Nightly, or you might be doing this in a released version where you've changed security.tls.version.min yourself.)