Weight is a value that can put a higher priority on an article and make it display ahead of newer content.
It is located below the 'Publish' button on the copy flow column on your articles in gryphon.
Normally, articles are setup to display newest first, but if you gave one of those articles a weight of '1', it would display first even if you published a newer article.
Weight values are from 0 to 5 with 5 being the highest importance.
Weight is setup to diminish by 1 point per day after you published the article. That is to filter weighted content back to normal after a period of higher importance.