In an older version of the flagging system, flags were handled by two main queues. The diamond moderators, obviously, could see and act on flags at will; that's one of the main jobs of being a diamond moderator. However, users with sufficiently high (10,000) reputation could also see many sorts of flags, but had only a lesser ability to act on them.
The privileges available at 10k reputation were (and still are) sufficient to do many community moderation tasks; for example, the ability to delete questions could usefully be used in response to valid flags on the question. Even lesser privileges could be very helpful: raising additional flags on a particularly problematic post would prioritise the post within the moderator queue. However, most of the community moderation abilities are intended for content that needs to be changed in some way.
As such, if a flag was nonsensical or invalid, a separate method was needed (and thus provided) for a 10k user to help in the flagging process; they could flag a flag as invalid (here's a question discussing when to do that). This was, at the time, the method via which a flag became disputed. Disputing a flag had no direct impact on the flag or the post, but a moderator reviewing the flag would be aware that there was a disagreement.
The 10k flag queue was considered problematic, so the moderation end of the flagging system was changed fairly comprehensively in April 2014. Instead of going into a 10k flag queue, flags that can potentially be handled without diamond moderator powers (such as "not an answer" flags and "very low quality" flags) now send a post into a review queue, with the 10k queue no longer existing (the rules for diamond moderators didn't change; they can still see and act on flags if they wish); and instead of being marked disputed when receiving an "invalid flag" flag, flags will instead be marked disputed when an action is performed that contradicts the flag (for example, a "Looks OK" review outcome in the Low Quality Posts review queue will dispute a "very low quality" flag, as the actions take entirely opposed viewpoints on the post). As a result, "invalid flag" flags no longer exist, and posts discussing them are typically of historical interest only.