When a bug is discovered in MediaWiki, or when a feature request is made, it spawns a lifecycle which involves several processes, including reporting, fixing/implementation, testing and release.
We rely heavily on feedback from users of MediaWiki, particularly users of Wikimedia web sites, who will often be the first to notice an issue between releases; third parties tend to notice longer-lasting issues.
A bug report or feature request exists from the moment an appropriate bug is opened on our BugZilla installation; http://bugzilla.wikimedia.org. Users are asked to provide a summary of the bug, some details about the version of the software in which it was encountered, and provide a full description of what the problem is, what they were doing, and what they expected to happen; in the case of feature requests, this is replaced by a description of the desired feature's behaviour.
Once a report is filed, emails are sent to several core developers, as well as subscribers to the wikibugs-l mailing list, which tracks changes in bug status, etc. Incoming bugs are often triaged and various properties set or adjusted, e.g. priority, severity, summary clarifications, etc.
Essentially, within about 48 hours of the bug's filing, it will be assessed as to whether or not the bug is in fact a bug, or whether a feature request will be implemented. At this time, non-bugs are usually closed as INVALID, while feature requests may be closed WONTFIX.
In the case of a true bug, the next step is to identify the cause.