CyberFaction

The Faction was a particular social group of Cyberman. They stood in contrast to Mondans. Both groups, however, originated on the planet Mondas and were, in a broader sense, "Mondasian".

Whereas the Mondans wished to retain some connection to their biological form, the Faction saw greater logic in fully embracing cybernetics. They thus retained a smaller percentage of their organic systems than Mondans.

The Faction were also opposed to the plan of the Mondans to pilot Mondas back to Earth to rob their twin planet of its energy. When that plan failed, and Mondas exploded, (TV: The Tenth Planet) the Faction were the only group of Mondasian Cybermen left in existence. They emerged as the "victorious" branch of the Cyber race. (PROSE: Iceberg)

Throughout his lives, the Doctor mostly encountered descendants of the Faction. However, neither he nor anyone else he encountered, regularly used the term "the Faction".

Behind the scenes
The term "the Faction" has its origins in the non-narrative encyclopaedic book, Doctor Who: Cybermen, by David Banks. Among the terms and theories invented by Banks to explain the variations in the costumes (worn by Banks and his fellow Cyber-performers over the years) was that of "the CyberFaction". None of Banks's proposed terms garnered much attention amongst DWU authors, and weren't carried forward in narratives until Banks's own novel Iceberg.

While Iceberg did introduce many of Banks's earlier-stated concepts into Doctor Who fiction proper, it is worth noting that even he failed to use his own coinage ‘CyberFaction’ in said novel, with the Cybermen in question being only referred to as ‘the Faction’. Even then, Banks remained the only author ever to speak of “the Faction”, for all that his treatment of said Faction implies that a vast majority of the Cybermen encountered by the Doctor up until then had belonged to said Faction.

Other Faction-related facts from Doctor Who: Cybermen not reinstated anywhere in fiction include the idea that after the destruction of Mondas, parts of the Faction became CyberNomads and left Sol to explore Mutter's Spiral; and that following the attack on the Gravitron, the Faction abandoned the Sol system and journey into the galaxy to colonise a suitable planet, eventually landing on Telos, which they colonised in 2175, resulting in ‘CyberTelosians’,  thus accounting for the apparently-native Telos Cybermen in The Tomb of the Cybermen in the same way suggested by Attack of the Cybermen three years prior — namely, that the Cybermen in question had actually been the results of Mondasian Cybermen taking over the planet.

Neither of these events are taken into account by this Wiki, as Doctor Who: Cybermen, being non-narrative, is not considered a valid source here.