So first off, let's answer the explicit question; What Does the Clan Ring Do?
The short answer?
The Clan Ring Does Everything
The Clan Ring is essentially the government of your clan. In addition to advising you as you make decisions and manage your tula, the success or failure of any action you take is determined by your clan ring, both directly and indirectly. This happens in a number of ways, but at the most basic level, the stats of your ring determine the likelihood of the success of everything your clan does. In some cases, this will be based on a specific member of the ring. In other cases, it will be based on the gods they worship. In still others, it might be random. Regardless, the members of your ring are the primary factor determining whether your clan is successful in Dragon Pass.
The first, and most obvious impact of your clan ring comes in the apportionment of clan magic. Having worshippers of 'war gods' on your ring (Orlanth, Elmal, Humakt, Urox, Vinga) will increase the number of points you are able to spend each year on War magic. Similarly, placing a Chalana Arroy worshipper on the ring will let you allocate an additional point to Healing. Eurmali offer you the chance to spend extra on Quests. Uraldans on herds, and so on and so forth.
Additionally, your clan magic is stronger (i.e. you have more points to allocate) when your ring is balanced evenly between men and women (4/3, either split is fine). Similarly, your magic is stronger when your ring consists of worshippers of seven distinct deities; additionally, heroquests are more likely to be successful when seven gods are represented on your ring.
Up to three members of your ring may be specialists, fulfilling an important role in your clan. You're obviously aware of the fact that you have a Clan Leader, but it can be easy to lose track of your Lorespeaker and War Leader unless you have those roles in mind.
Clan Leader: Occupying the leftmost slot on the ring, and denoted by the Torc of Leadership, the Clan Leader is the only member of the ring whom you can directly appoint. Ideally, this should be the member of your clan with the highest leadership score, who worships the same god as your clan. That means an Elmali for an Elmali clan, an Ernaldan for an Ernaldan clan, and either an Orlanth or Vinga worshipper for an Orlanthi clan. While it's possible to appoint a clan leader with a low leadership score, or a worshipper of another god, be advised that eventually your people will become upset by this, and you could have a coup on your hands. Eurmali make particularly bad chiefs, although it's definitely fun to try to keep them in power for as long as possible. As you approach the endgame, stats beyond Leadership, most notably Magic, Combat, and Custom, also become quite important.
War Leader: The warleader is the member of the ring with the highest Combat statistic, and usually ends up occupying the slot immediately to the right of the Clan Leader. Ideally, he'll have strong Leadership and/or Magic too. While it's possible for your Clan Leader to pull double duty in this role, it isn't always ideal; worshippers of war gods like Humakt, Urox, and Vinga(Or Orlanthi, but remember, avoid redundancy!) excel in this role (Especially Humakti and Vingans who can be easily groomed for the role through heroquests). This role is automatically assigned. You don't get to choose who your warleader is, though since the criteria is quite simple, it's fairly easy to force the role onto somebody. Your warleaders Combat skill is one of the primary determining factors in the success of your clan in both offensive raids and defending your tula against attacks. He or she may also be called upon in a variety of other special events.
Lawspeaker: The Lawspeaker is the member of your ring who is going to be responsible for providing you with good advice when it comes to matters of law and tradition, the resolution of disputes, and the forging (and dissolving!) of alliances. In general, the Lawspeaker will be the member of the ring with the highest Custom score who is not your clan leader, and, ideally, will be a worshipper of Lankhor Mhy. In fact, a Lankhor Mhy worshipper will even usurp the role from a worshipper of another god with a higher custom score!
In addition to the three formal roles on the ring, there are also a few informal roles that you'll probably want to fill in order to maximize the success of your clan. You'll often end up doing without some of these - either by choice or necessity, and many of these can be doubled up with other roles. For instance, I often find that my Lawspeaker ends up being my magician, though that's not ideal.
Magician: You want somebody on your clan ring with a high Magic stat. Full stop. Magic is relied upon to determine your chance of success or failure in more events than any other single stat (Excepting, I suppose, Combat if you include raids), so you probably want the member of your clan with the highest magic available on your ring.
Trickster: Worshippers of Eurmal open up a variety of alternate resolutions to events, and protect you from funny looking cows and ravenous armadillo infestations. In addition they allow you to place more magic into Quests in years in which you intend to Heroquest and generally improving your odds of success on the Godplane. Of course, keep in mind that Tricksters can be as much a curse as a blessing, so YMMV.
Bargainer: Ideally a worshipper of Issaries, you'll want someone on your ring with high Bargaining skill to negotiate with traders and emissaries that show up on your doorstep. This may not however, always be the person with the highest Bargaining in your clan. (See 'Leaving the Tula' below for more on this subject).
Farmers: You'll want a ring member with high Plants and Animals if possible. This tends to be my lowest priority in forming a ring, but is generally easily satisfied simply by checking off all of the other boxes you need to in the construction of a balanced ring. These stats will affect the fertility of your herds, the bounty (or lack thereof!) of your harvests, and your ability to handle agricultural events such as droughts or cattle disease.
Banner Meat: There is one other role that I often find myself assigning on the clan ring, and that is a useless noble whom I can place on the ring to kill with Humakt's Raven Banner. The Banner guarantees that you will be victorious in any combat in which you use it, but whoever carries the banner is guaranteed to die. While normally a Weaponthane will do so, occasionally they will refuse and demand that a noble carry it. Having an easily replaceable noble on the ring to kill off in this manner is handy. This is also a nice way to force the game to generate new nobles, if, for instance, your Lawspeaker is getting on in years and you have no young Lankhor Mhy worshippers, or you need a new Trickster, etc. Barntar worshippers with decent Plants/Animals skills are particularly abundant and useful in this role.
Leaving the Tula
Occasionally, you may be called upon to send a member of your ring away, or you may choose to do so, on an exploration or trading mission. Be advised that if you do so, you will be unable to receive advice from that noble until they return, and they will not contribute to the success or failure of your actions while they are away. It is for this reason that you may not want your best bargainer on the Ring - a good Issarite can often do as much or more good on the road as they can on the Tula. This doesn't mean you should never send ring members away, but be very aware of the implications when you do.
Bringing it All Together
So, how does a typical 'good' ring look then? For an Orlanthi clan, an ideal ring might look something like this:
Clan Leader: Orlanthi, Priority stats are Leadership, Combat, Magic, and Custom
War Leader: Vingan, Priority stat is Combat. High Leadership and Magic a plus.
Lawspeaker: Lankhorite, Priority stat is Custom. High Magic and Bargaining are bonuses.
Trickster: Eurmal, could be of either gender. It'd be nice if they have high Magic, Plants, or Animals, since that'd be helpful, but that's not why you put a Trickster on the ring, now is it. (Though Trickster War Leaders and Lawspeakers can be fun!)
Bargainer: Issarite, Priority stat is Bargaining. High Custom is a plus.
Magician: Chalanist, highest Magic in the clan.
Farmer: Ernaldan, high Plants and Animals.
Attentive readers will note that the ring I've just posited has at least 2 men, and 2 women. Worshippers of Issaries, Chalana Arroy, Eurmal can be of either gender. So long as all three aren't the same, you'll have a gender balanced ring with 7 gods in this case, that allows you to place up to 2 extra magic in war, and one extra each in Crops, Trade, Health, and Diplomacy. Of course, doubling up some of the above roles - maybe your Trickster has high Plants, or maybe your Clan Leader can serve as War Leader, frees up slots for you to include worshippers of more exotic gods like Odyala, Urox, or Barntar. (Just kidding, there's nothing exotic or special about Barntarites. Aside from their expendability.)
The last thing to keep in mind is that, over time, nobles on the ring will see their stats improve, especially late in the game once you can allocate magic into Heroism. Furthermore, you can use Heroquests to specifically groom and train your nobles to fill key roles. One thing that's common, for example, is to send a young worshipper of your clan's patron on a number of Heroquests to raise their Leadership, Combat, and Magic stats to groom them to take over the leadership role as your current leader ages. You can also 'train' nobles by sending them away from the Tula on missions. Exploration missions can raise a nobles Combat and Leadership scores. Leading caravans will increase their Bargaining; and, if beset by bandits, possibly Combat and Leadership as well! Similarly, elderly nobles may see their stats decline, or of course, they can die! Having nobles of diverse ages on the ring can help you avoid a mass die-off and crisis of leadership.
In the end, you have a great deal of flexibility, so long as you keep in mind the key principles of filling the three formal roles, keeping gender balanced, and including a diversity of cults. Beyond that, feel free to simply stack your clan ring with whatever nobles have the coolest portraits or hate elves the most. And put the Trickster in charge once you've gotten the hang of the game. Everyone ought to, at least once.