Final answer:
The Illinois Constitution grants local governments more power through home rule, which allows for local discretion in governance within the limits of their charters.
Step-by-step explanation:
The feature in the Illinois Constitution that gives various units of local government more power in governing their own affairs is known as home rule. This autonomy allows local governments the discretion to make policy or tax decisions within a charter that outlines local government responsibilities and areas of authority. The concept of home rule is a transfer of power aimed to better cater to the needs of individual communities, though potential conflicts can occur if state governments choose to override local decisions, as seen with the example of the State of Texas overruling a fracking ban imposed by the City of Denton.