Awards are what you get for completing certain thresholds of harvesting plants. To access the awards menu just go to the Quest Menu and click the tab on the top right that says "Awards". This will bring you to the Awards menu!
You can see Awards you are currently making progress on, and awards you have mastered. To get additional details just click the Question mark on the award square.
This will tell you the name of the award, your progress (how many plants you've harvested and how many more to harvest) and the reward when you complete that particular award.