For global events, you need to be a site administrator. You will then see the corresponding option when creating a new event.
For group events, you must be creating an event in a course with groups enabled, and be an editing teacher in that course. You will then be able to make a group event.
After you create a group event, the calendar never lets you see to which group the event belongs, nor does it let you change the group to which the event belongs. Is this a known bug, or am I missing something perhaps? (I'm using Moodle 1.9.2).