Being a registered Society is one option, among many, for a student run group.
A Society qualifies as a student-led group, who form organically to perform activities of shared interest.
Edinburgh University Societies must:

Focus on Edinburgh University Students

Enhance the student experience

Be autonomous groups

Follow the structure as outlined in the Students' Association Society Constitution.
They must have long term goals and plans.
There must be no distribution of profits made.
Not overlap with any Societies that already exist.
You must look at the Proposed Societies' Policy to decide what category you would like to apply in.