To assign is to specify something or someone for a specific purpose. If your sisters fight over whose turn it is to sit in the front seat, your parents may have to assign turns.
The verb assign also means to transfer legal rights. You assign the deed to your house to new owners when you sell it. If you sort things into categories by characteristics, you are assigning them categories. When you make a budget, you assign each expense a category. Food and housing, for example, may be essential living expenses and medicine and doctor's visits may be health care. Chocolate gets its own category.