Hey there. Not a stupid question at all.

Was it the tribe/thing that made Muhammad flee from Mecca to Medina?

Muhammad was forced to flee Mecca by powerful members of his own tribe, Quraysh. That tribe had a number of clans, subdivisions of the tribe based on a common ancestor.

The Umayyads were one of those clans, part of the Banu Umayya, which is to say that they were all descendants of Umayya ibn Abd Shams. Abd Shams was Muhammad's great-grand-uncle. (Note, the "-ad" in "Umayyad" is a suffix like the "-id" in "Abbasid" or the "-ite" in "Hashemite." It denotes descent from Umayya, Abbas, and Hashem, respectively.)

The Umayyad's rise to power over the newly emerged Arab/Islamic empire began after Muhammad's death. The third of the four "Rightly Guided" caliphs (the "Rashidun"), Uthman ibn Affan, was of the Banu Umayya.

After Uthman's assassination, there was a succession crisis between Ali, the Fourth Caliph, and Muawiya, Uthman's second cousin.

Muawiya is usually reckoned as the first Umayyad Caliph because Uthman is usually counted as one of the four Rashidun.

The empire that Muawiya presided over was already incredibly expansive at this point, but it was under the Umayyads that it would reach its maximum extent, stretching, as you say, from Spain to India.

The Umayyads ruled for about 100 years before they were deposed in a civil war by a rival clan, the Abbasids, in 750.

Virtually every member of the Umayyad clan was killed, but those that survived fled to Spain. Abbasid rule was characterized by the breakdown of the unitary Islamic empire. What had been a single state came to be ruled successor kingdoms, even though the Abassids themselves continued to rule territory. Those Umayyads that survived in Spain established one of these successor kingdoms, the Emirate of Cordoba. In the 10th century they went one step further and declared themselves Caliphs of Cordoba. This was in a turbulent period in Islamic history when there were multiple rival Caliphates and the overall power of the Abassid Caliphate was in sharp decline.

Edit: Also worth noting, it seems as though the wikipedia entries on the Banu Ummaya and the Ummayads have all been edited to reflect a Shia sectarian line that Umayya was the adopted son of Abd Shams and not a blood relative of Muhammad at all. This is not a notion that I've ever come across, and having seen these entries a few hours ago I'm failing to find any reputable source to support it. In general, Wikipedia entries on Islam and Islamic history are an absolute mess. They often reflect deeply sectarian information, or present only the devoutly religious, non-academic, non-critical perspective.