'Back in the day' is a pretty vague, but I'll give it a shot. There are many recorded tribal names, but I'll stick to the more significant ones.

Prior to recorded history, we think people in Britain were speaking a language related to Basque, but we don't have any records of this language. As Celtic language and culture spread out from Central Europe, they made their way into Britain and Ireland. When we start getting recorded history, both islands were mostly Celtic. Though the Picts may have not been Celtic, we don't really know. These people were tribal and some of them had close ties with tribes on the continent. The Belgae, for instance, were in both Southern Britain and the continent. They gave their name to Belgium. These tribes would have had kings or chieftains and occasionally formed alliances against greater enemies, like Rome. The Gauls, led by Vercingetorix, were really a confederation of the various tribes in the region. Caesar's war in Gaul was, in part, due to Roman alliances with various tribes (like the Helvetii, whose name survives in a font: Helvetica), and the Gaulish war effort was aided by Belgae in Britain. Here's a list of known Celtic tribes: http://en.wikipedia.org/wiki/List_of_Celtic_tribes

When the Romans invaded, they weren't able to fully romanize Southern Britain before they withdrew. They managed to build some cities, and Christianity took hold. The Iceni are a tribe who famously revolted against Roman rule under their queen, Boudica. Ireland was never part of Rome, but adopted Christianity fairly early and Irish missionaries converted the Picts in what is today Scotland. A group called the Scotti spread into Scotland, bring the Irish language with them, and historical sources often used the term 'scot' to refer to the Irish, or any Gaelic speaker. The Irish colonized a region in Scotland called Dal Riata, and they eventually formed a united kingdom with the Picts.

After the Romans left Britain, the people reverted to tribalism and abandoned the cities. Chieftains took over. The Roman general Magnus Maximus, who withdraw to Gaul, shows up in Welsh folklore as the guy who gave Wales back to the Welsh chieftains and he's regarded as something of a hero-figure.

Pagan Germanic peoples from the continent began setting up small kingdoms in what was to become England after either being invited in to fight as mercenaries or just to fill the power-vacuum left by Rome. For more on this, look into the Anglo-Saxon Heptarchy. Whether or not they killed and drove out the previous inhabitants or anglicized them is debated, but genetic studies seem to support the idea that the population genetics remained the same. These kingdoms were converted by Frankish and Irish missions. Over time, the various Kingdoms consolidated, but they has their roots in continental tribes: the Angles, Saxons, Jutes, and Frisians.

Tribes in Wales gave way to kingdoms like Gwynnedd and Dyfed. Gwynnedd is sometimes considered a Roman successor state and takes its name from the Venedoti tribe. Dyfed comes from the Demetae.

Sometimes there's a big blurry grey line between tribe and petty kingdom, but most of the tribal groups in Britain either became unified through conquest or confederation.