The chart was a ranking system the Watterson's used to gauge the value of their accomplishments throughout the week. There were apparently five spots. They are (from bottom to top): rainy cloud, gray cloud, sunny cloud, sun, and star.
The chart plays a major role in "The Fridge", where the Watterson family obsesses over getting the star spot. Unfortunately, Nicole is the sole decider of what ranking each family member gets. Even though other family members accomplish great feats (such as Anais winning a prestigious science award), she refuses to give them anything higher than the sunny cloud. This drives the family mad, and culminates in a viciously competitive free-for-all paintball match.
Gumball appears to be the only one that does not obsess over his ranking on the chart. Nicole is troubled by this, and attempts to train him (rather extensively) to value winning over everything else. She eventually awards him the sun spot; Gumball feels proud of himself for this achievement, but not to the extent Nicole was hoping for - it was not enough to inspire him to sacrifice anything and everything for the taste of victory. He even refuses to take part in the paintball match at the end, but agrees when Nicole offers to take down the chart if he wins. Gumball eventually does win, and the chart goes down.