Where does "flag weight" come from and what does it mean? Some of the badge descriptors in general are sort of ambiguous in their description because the use terminology not familiar to the common user.
If you flag a post (using the 'flag' link below it) the moderators get notified. Any moderator can either dismiss your flag as valid or invalid. Some moderator actions automatically dismiss your flag as valid, like if the moderator closes a post flagged as off-topic by you. For every valid flag you gain 10 flag weight and for every invalid flag you lose 10.
You start with a flag weight of 100 and once you reach 500 (the original maximum) you get the silver deputy badge. The higher you flag weight is the more flags you get per day and the higher your flags are ranked, i.e. your flags will be listed before flags of people with lower flag weight. For smaller sites with few flags this isn't important but for e.g. the stackoverflow.com site there might be hundreds of flags at the same time and this technique allows the moderators to process the good flags from people which have proven to flag correctly first.