Responsible disclosure is the process where you as a maintainer receives a report about a security problem and work on a fix privately until you have a fix and then disclose the problem to the world coordinated with the fix.
A responsible disclose can also often involve pre-notifying distributors or vendors that ship your product so that they can be prepared and offer fixed versions to users on the same day you make the security problem known to everyone.
I am a strong proponent of the responsible disclosure approach because of how it tries to keep innocent users of the product safe. As soon as the security flaw becomes known, we can be sure that malicious actors will try to take advantage of it for nefarious purposes.
This said, the keeping things private for the sake of the users' safety must not be abused or taken lightly. It should only ever be used for actual security problems and not for any other kind of bugs. It is also important that the problem still gets fixed and gets published within a reasonable time even when handled in private. Because a security problem can of course still be exploited and hurt users even before you have announced it to the world.
Critics of this model tend to favor shipping the bugfix sooner rather than later in order to help minimize the time window for which bad guys can abuse the issue. That allows users to patch their systems sooner, but might also leave users who cannot patch their own systems (for whatever reason) vulnerable for a now public flaw for a longer period of time.