What are campaign groups? Do I have to create one in order to create a campaign?

Yes, a group is needed to place the campaign in. Campaign groups are like folders in a computer made to help you sort your campaigns. You can also set daily and total budgets for the group itself, which will apply for all the campaigns within that specific group. You can not move campaigns between groups, but you can clone them and place the clone in another group.

All campaigns belong to one campaign group, and one only. Moreover it is not possible to move an existing campaign from one group to another.

It is possible, though, to create a clone of an existing campaign and put it in any other group.