It isn't everything: only the Regent can directly command another member of her Chantry to act. However, woe unto low ranked Apprentice who disobeys so much as a hint from the Chantry's Primogen. And if the Regent has assigned you as assistant to another's task, obey that person!
Rank governs a lot of the public things within the Clan. Tremere will enter an official Elysium meeting in Rank order, and leave in Rank order. If two Tremere meet publically, the lower ranked gives deference to the higher.
It also governs private matters. And effective rank inside the confines of the Chantry walls may differ from nominal rank a lower level Apprentice who repeatedly achieves for the Tremere may earn more respect than an idle higher level Magus.
Rank can be affected by politics and by one's activities on behalf of the clan as much as by Thaumaturgic ability.
Within a Chantry:
Apprentice of the First Circle
Usually a new Tremere. Demotion to the First Circle would be a hideously embarrassing event for an older Tremere.
An Apprentice of the First Circle is usually under strict oversight; reporting to the Regent on a weekly basis, frequently living in the Chantry, and often being given assigned tasks. Most apprentices of the First Circle are studying their first rituals, learning the basics of Thaumaturgy (almost always the Path of Blood), and learning how to be Kindred.
Once an Apprentice of the First Circle has mastered the basics of being Kindred and Tremere, understands the Oath and the Traditions and can be counted on not to egregiously breach the Masquerade or embarass the Clan, they are promoted.
Apprentice of the Second/Third Circle
Apprentices of the Second and Third Circles are still studying the basics. They are learning the deeper meanings of the Traditions and the Oath, learning the basic Occult Principles, and proving their ability to work without direct instruction and supervision.
Think of the First Circle as being more like school, and the Second and Third as like college/university. In the First Circle, you're fed information and given a timetable for learning it. In the Second and Third Circles, you're told what you need to learn, given the resources (including teachers) to learn it, and left more to your own devices.
Depending on how busy the Regent is, Apprentices of the Second and Third Circles may need to make weekly reports, or monthly, or somewhere in between. They must still report regularly. Some Regents prefer that they live in the Chantry, others permit them to have Havens elsewhere.
Apprentice of the Fourth/Fifth Circle
At the Fourth and Fifth Circles, advancement comes more slowly. The basics have been learned, and the Apprentice has proven themselves to be able to function on their own. Now the more senior Apprentices and the Regent give him or her tasks and leave them to it. They're rarely told how to do things, instead they have to figure it out.
They also have time for personal projects, and are encouraged to pursue them. Study materials for Thaumaturgy and other Occult matters are readily available, and the Apprentice is expected to use them, but noone is going to quiz them on it.
They may reside in the Chantry if they wish, but they don't need to. They must attend the weekly Chantry meeting and the yearly all Tremere meeting near Halloween; but their reports to the Regent may be as infrequent as once a year.
(Side note: Apprentices of the Fifth Circle and above may choose to use the title 'High Apprentice'.)
Apprentice of the Sixth/Seventh Circle
The Primogen and other senior members of the Chantry, other than the Regent, are Apprentices of this level. At this point, Apprentices are simply waiting for a new Regency to become available or are personality types who prefer not to be a Regent, Lord or Pontifex.
Newer Apprentices of this level are learning the duties of the highest ranked Tremere, and proving their usefulness as bureaucrats or proving that the Clan is better served keeping them as researchers, diplomats, or in other roles.
Older Apprentices of this level are usually those who serve the clan best in those other roles.
Apprentices at this level have more freedom than any other in the Clan. A Regent, Lord, or higher has the responsibility of caring for those below them: these Apprentices are required only to serve the Clan in some way, attend the compulsory meetings, and make irregular reports to their Regent.
Regent
The Regent runs the Chantry and is the administrative head of the Chantry. He or she is not necessarily the oldest or the most powerful Tremere in the Chantry, but is an older and experienced Tremere who is suited to the demands of the position.
Beyond the Chantry:
Attaches
Apprentices of the Fourth Rank or higher may become a personal assistant - or attache - of a Lord, Pontifex, or even the Council of Seven. Such Apprentices don't technically belong to a Chantry, but work directly for their senior.
Lord
A Lord manages multiple Chantries over a geographic area. He or she is responsible for all occurances within the area, and for ensuring that the Clan flourishes regardless of what events occur there.
Pontifices
A Pontifex (plural: Pontifices) is responsible for a broad geographic region; such as a continent, or 'the South Pacific Islands'.
Council of Seven
The Council of Seven rules all of the Clan. End of story.