"House Military" was the term for the collective armed forces of the Great Houses during the War.
The House Military was not a cohesive and official unit, because (supposedly) all of the Houses were equally dedicated to the War effort. Instead, the term referred to a loose alliance of Houses (notably Arpexia, Xianthellipse, and Mirraflex) and their military presence in the Spiral Politic outside the Homeworld. (PROSE: The Book of the War)
History
Pre-War
Before the War, the Great Houses' only military force was their ceremonial guard. This ceremonial guard was monstrously ineffective, proving itself completely useless in repelling the (ultimately doomed) invasion attempts of aliens (PROSE: The Book of the War) like the Sontarans (TV: The Invasion of Time) and Daleks. (AUDIO: The Apocalypse Element)
The one exception to this was the House Military formed under the Imperator Presidency. It was formed from lesser species recruits as well as the Imperator's followers, and created havoc across the Spiral Politic until the Imperator's execution. (PROSE: The Book of the War, The Return of the King)
Waves
Instead of dividing their military into legions, or by purpose, the Great Houses primarily organised the House Military into a sequence of generations called Waves. The War took the form of a cycle between the instantaneous deployment of an entire Wave, and a long lull of entrenchment as the Houses concentrated their effort on the design and gathering of biodata for the next. Each Wave tended to curb its predecessor's excesses, while emphasising the traits that had lead to success. Instead of numbering campaigns in order, the Houses numbered them based on Wave: the Third Retro-War was named for the Wave involved, not for the number of Retro-Wars preceding it. (PROSE: The Book of the War)
The First Wave of the House Military was comprised of normal House members, many originally part of the ceremonial guard. They were the only Wave not designed specifically for War, although some were fitted with the few militarily-applicable augmentations available at the time. The First Wave only met with any success because their enemy's early forces were equally unready. Few to no First Wave soldiers who fought in the War's first battle, on Dronid, survived. (PROSE: The Book of the War)
The Second Wave was the first created specifically for war, by a Homeworld with only received knowledge of what war entailed. It was extreme by any measure, a crusade which committed frequent acts of genocide, wiping out any species or culture which opposed — or even rivalled — the Homeworld. Its comprehensive destruction of Faction Paradox sites like Ordifica crippled the Faction for years, and ruined the lines of communication between the Faction and the Remote — leaving the Remote unbound by Linearity, and ultimately resulting in Compassion's creation. (PROSE: The Book of the War)
Third Wave troops were created in hominid forms, but they became non-hominid with each life to a far greater extent than the Second Wave. (PROSE: The Book of the War)
House Xianthellipse took initiative with the Fourth Wave, introducing foreign biodata (including that of humanity) to the breeding engines (much to House Mirraflex's chagrin). Compared to the Homeworld's traditional values, the Fourth Wave was chaotic — its soldiers were undisciplined (best suited to guerrilla warfare), tended to prod the boundaries of the Houses' Protocols, and even had sexual tendencies. Robert Scarrat was the most high-profile (and arguably most successful) of the Fourth Wave. (PROSE: The Book of the War)
The Fifth Wave was stable and standardised. The Houses incorporated the military cell-codes of the Second and Third Waves and the "mongrel" biodata techniques of the Fourth Wave, but were more careful to keep those Waves' more extreme tendencies under control. It was the first Wave to be created without any political drama. (PROSE: The Book of the War)
Officially, no Sixth Wave ever existed — the number was simply skipped. However, there was a suspiciously long gap between the Fifth and Seventh Waves, and The Book of the War referred to a non-existent entry on "Sixth Wave Defections," apparently expunged by the enemy. (PROSE: The Book of the War)
The Seventh Wave was created when Fifth Wave troops were still plentiful, so rather than a general fighting force, it was used for specific tactical methods. It was overseen by House Xianthellipse, and combined the savagery of the Second Wave with the subtler, less culturally acceptable techniques of the Fourth to psychologically degrade its targets. The Seventh Wave was used in the Seventh Time Assault, a "morally questionable" campaign which took place on Earth and was inspired by apportation. After the enemy became involved, the Assault eventually (or inevitably) devolved into the Seventh Retro-War. (PROSE: The Book of the War)
The Eighth Wave was defensive, designed only to reinforce the troops already in use. (PROSE: The Book of the War)
The Ninth Wave was odd in multiple ways: the non-military House Tracolix was instrumental in its creation, even though it had never previously been involved in breeding, and claimed a "surprise" was in the works. (PROSE: The Book of the War) It could have had something to do with House Lolita, which was allied with Tracolix (PROSE: The Book of the War) and the only member of which (Lolita) was a timeship (PROSE: Toy Story) who spread her own children throughout history. (AUDIO: In the Year of the Cat)
Minor references
Tonton Macoute once cooked with the bodies of House Military soldiers. (PROSE: Tonton Macoute)
The House Military were sent to the Great Houses' prison planet to stop a prison break orchestrated by a posthuman from Siloportem, Demetra Kein. (AUDIO: A Labyrinth of Histories)
Biology
Because the First Wave was composed of ordinary House members, its troops were all hominid. But when Second Wave troops were killed, their new bodies would be slightly less hominid, with whatever flaw resulted in their death removed. The idea was taken farther in the Third Wave, with regen-inf soldiers who became living siege engines, with built-in temporal senses, defences, and weaponry. (PROSE: The Book of the War)
Culture
There was an unofficial cultural schism (exemplified by the Redemption Cult) between the House Military and the Great Houses proper back on the Homeworld. House Military troops often saw themselves as the only active upholders of the Homeworld's laws and culture, looking down on the Council of the Ruling Houses and even their own time strategists for only acting from a distance. The schism was also encouraged by the soldiers' biological nature: forced regen missions created a death-fetishism more natural to Faction Paradox than Homeworld tradition, and the physical incongruity between sheltered hominids and the living tanks of the House Military only grew with time. (PROSE: The Book of the War)