So, this is something that has been bothering me for a while, in various contexts when discussing arab history (that is, history of the arabian peninsula) the concept of tribes pop up, be it for the ruling dynasties of kingdoms (eg. the Ghassanids), or when discussing the the prophet (pbuh) Muhammed and him coming from the tribe of Quraysh. Problem is, a lot of the time various secondary sources seem to assume that one is already familiar with how these tribes worked, and what they were, and refer to them kind of assuming the reader is already familiar with the concept.
The basic gist seems to be a set of family groupings deriving from a common ancestor (possibly mythological) but what kind of social and political role did they play? There are sometimes alluded to subdivisions?
So, I'm re-reading Albert Hourani at the moment, and your definition is relatively spot on: a tribe is an extended family group. Why "tribe" became the accepted term in English instead of, say, "clan," (which IMO would be clearer), I could only speculate about--the term in Arabic, qabila, can be translated as either.
Hourani discusses this in the beginning of Chapter 1 of A History of the Arab Peoples; in the fairly common Warner Books paperback reprint it's on pages 10-11.
As far as I am able to tell, most of the tribal ancestors were usually real people (as far as we can attest for this period). Knowledge of genealogy was quite important and inserting a fictitious person into the mix would have been easily detected.
The tribal system was important because Arabia was almost never under centralized rule in the pre-Islamic period, so the resources of the peninsula came under the control of specific tribes, who had to negotiate with others how to access or share them. This would include grazing lands for livestock, wells, the fertile land in oases, access to market cities controlled by other tribes, etc. So, the tribal system allowed for a decentralized system that functioned as long as everyone knew who everyone else was, which is why identifying one's self by their lineage was so important.
Areas of control might be subjected to power sharing arrangements--we get this oasis in the fall, you get it in the spring--and terms were constantly shifting, which is why you don't see maps with firm borders on them but rather approximate areas.
Of the pre-Islamic tribes, the three most well-known are the Quraysh (controllers of Mecca, Muhammad's tribe), and the Lakhmids and Ghassanids because the latter two were on-again off-again client-proxies for the Sassanids and Byzantines, respectively. For those who study pre-Islamic Arabia, or the history of the Arabic language, the intricacies of the tribal divisions are important because tribes often had their own dialect of Arabic, controlled certain lands, etc.
The most important takeaway for the early Islamic period is understanding that tribal identity was extremely important--a man without a tribe was essentially without identity or any social standing. Muhammad and the early believers tried very hard to erase these tribal identities, but even then they weren't entirely successful. There were early struggles between Quraysh and non-Quraysh, and then later on the privileging of members of the Umayyad family over others when the capital was moved to Damascus (along with the privileging of Arabs over non-Arabs). These were all issues that the early community had to work out--and, even later on, those who could trace descent to one of the original Arabian tribes that joined Muhammad's movement were often treated with privilege and deference, even though Muhammad himself is said to have discouraged such behavior.
Another excellent source that gets into the intricacies of this is Marshall Hogdson's The Venture of Islam, Vol 1, Chapter II, Muhammad's Challenge. He is a bit longwinded, but if you're looking for detail, this is where you'll find it.