Ability Management is a menu that allows you to customize your Capital Ship's Abilities by upgrading them. Clicking on it will give you the option to upgrade any of the ship's four abilities that are available for upgrade. It also allows you to buy training for your ship instead of working through it by destroying ships. Further research allows you to train to higher and higher levels. If you right click the ability management icon you can toggle automatically upgrading whichever ability the AI chooses, if you are willing to trust an AI to make wise long term decisions.