Today list contains tasks that should be done during the current day.
The list is displayed for each Project and Area and for People. Personal Today list contains tasks assigned to the current user from corresponding Project or Area-level lists.

For tasks from Today list assigned to the current user there is a start/stop button displayed that allows you to mark the task you are currently working on.
If there is a task scheduled for the exact date (and thus displayed in Scheduled list, such task is automatically moved to Today list when the date comes in Project or Area time zone (see Project/Area settings).