A group exists only while there is at least one record still using that group. To remove it, you need to either delete the records in that group or edit them to put them in a different group instead.
As soon as you edit the last remaining item that was stored as a member of that group, the group will automatically disappear.
In the case of Appointments and Tasks, keep in mind that the main screen is focused on showing you current things, not a complete list of every Appointment or Task you have ever had. If you are having trouble removing an existing group in Appointments or Tasks, it is usually an old or future item that remains in the group, forcing it to stay around. Instead of using the main screen to find and edit Appointments and Tasks, you can go to the Report area and search for any items of any age that were in the group you are trying to remove.
This help article applies to Intellect 3