Downtime activities are tasks that usually take a workweek (5 days) or longer to perform, and are often done inbetween adventures and typically done in populated areas where resources and equipment exist. These tasks can include crafting items, research and study, working, relaxing, buying/selling magic items, pit fighting, crime and more. Some acitivites require a monetary investment and a skill check, and not all tasks result in success.
The DM will have final say on which activities are available to the characters. The activities the DM allows might depend on the nature of the area where the characters are located. For example, the DM might disallow the creation of magic items or decide that the characters are in a town that is too isolated from major markets for them to buy such items.