What kind of military would you need to wage a war in space? You've got your Napoleonic space navies, which can lay waste to one another in empty sections of space. You've got space marines, who are armored up and ready to kill anything that moves. And finally, you've got the fighter jocks, who run interference and find those vulnerable, womp rat-sized holes to exploit. But are these representations actually realistic?
Militaries are very specific organizations that are created through quite a bit of trial and error throughout the years: they're the product of a collective and continuing history that has shaped their every aspect to what they are today. These organizations also change based on who they're forced up against: Look at the buildup of armies in Europe throughout the 1800s, or the disposition of the NATO and Soviet Union forces throughout the heights of the Cold War. If humanity expands into the cosmos, we'll undoubtedly bring our guns with us, and the disposition of the armed forces will reflect their surroundings, as well as take their history along with them.
Here on planet Earth, the various branches of our militaries are generally technology driven. An Army is a branch that "conducts military operations on land." A Navy conducts their operations at sea, while an Air Force conducts aerial operations. The Marines are trained to fight on land or sea. The Coast Guard is a little more nebulous, ranging from defensive military operations to peacetime security and law enforcement. A quick look at Wikipedia points to a couple of other types: Norwegian Cyber Force is responsible for electronic warfare as of 2012, while the Russian Aerospace Defence Forces are responsible for the nation's missile defenses, covering attacks from the air and from space. Russia also has the Strategic Missile Troops that focuses on ICBM deployment (in the US, this is handled by the Air Force).
So, how do you build an army in space, when the general military types have evolved for use on just one planet?
With only one exception, the entirety of Earth's entire military history takes place on the planet Earth. While much of humanity's near earth orbit and spaceflight activities have evolved out of military operations, it's predominantly oriented towards the people who are still on the ground. This is largely - but not exclusively - represented in science fiction entertainment.
A hard-surface force would be roughly analogous to an army, and we can likely keep the title for any unit that hits the surface of a planet. But, the wide variety of surfaces that actually exist. Any surface that would be immediately and overwhelmingly fatal to someone landing on it can probably be considered avoidable for military purposes, but the range that remains, could considered for combat operations. Any space-borne military would likely have a range of groups under the larger 'Army' umbrella that could respond to a comparable range of environments.
Take a look at our solar system for an example: Planetary bodies are just one type of hard surface on which an army could fight upon. In addition to the four rocky planets, here's a huge number of moons, asteroids, and planetesimals, each widely different. A soldier fitted and trained to fight on Earth would have a difficult time on Venus or Mars, where there's environmental and survival aspects on top of mission requirements. Surviving these worlds often comes down to equipment: specialized pressure suits, power armor and so forth are tools used to keep its soldiers alive. The same can be said for those on planet Earth. Just look at how much trouble American soldiers had during the Battle of the Bulge in 1944 when they went into battle in December with summer gear.
The sheer number of numbers of environments have been covered before. In Battlestar Galactica's Season 1 episode 'You Can't Go Home Again', we see Starbuck shot down on a very uninhabitable surface, one that she's clearly equipped to work on. Space: Above and Beyond also deploys its soldiers to hostile worlds. In the literary world, Joe Haldeman's novel The Forever War puts its soldiers into a wide range of planets with and without atmospheres.
Atmospheric & Flight Ops
Since the 1st World War, the invention of the airplane has created its own unique branch of warfighting: the Air Force. Since its inception, airplanes have provided a whole host of roles on the battlefield, from tactical support of the soldiers on the ground, to strategic operations intended to deter enemy actions, to logistical and transportation operations. If soldiers are to be placed on the ground on an enemy planet, in all likelihood, some form of support would be desired. Soldiers needs to be brought from location to location, but can also take advantage of the added firepower that an aerial platform can provide.
Ultimately, this is a fairly limited resource that can be put to use. The first requirement, obviously, would be the need for an atmosphere, if you plan on going with aircraft. But, even then, the atmosphere needs to be just right: Mars has an atmospheric pressure of 0.087 psi, while Earth maintains a pressure of around 14.69 psi. Venus, on the other hand, has a whopping 1,330 psi on its surface. Deploying aircraft on each planet would require an entire field of equipment that essentially couldn't realistically be used anywhere else.
You might not need aircraft. The Lunar Excursion Modules of the Apollo missions were the first spacecraft expressly developed to land on a surface other than the face of the Earth, and if needed, could hover in place while its pilots found a suitable landing zone. The same principle could be applied for any range of bodies in a solar system, especially for asteroids and smaller moons, where the gravity is far lower than that of their planetary neighbors.
Soldiers on the ground face a number of unique challenges when deployed to a planet's surface: atmosphere, gravity, enemy soldiers. In space, you eliminate many of those unique challenges through a concentration of resources. If you're intending on sending soldiers around your region of space, they're going to need to be brought together, which opens up an entire range of activities that need to be put to use.
In the broader sense, there's a strategic element that could impede troop transportation operations, depending on how well any given planet or nation has set up their defenses. You can take a cue from the Cold War, where NATO (chiefly the US) and the Soviet Union set up a system of checks against one another. As competing space-faring nations work against one another, they're going to build in ways that provide the greatest advantage over the other. While space-based warfare has often been compared to naval warfare of the 1800s, (notably David Weber's Honor Harrington novels or the Imperial Fleet from Star Wars), the similarities only go so far. In deep space operations, spaceships would need to have the ability to operate independently of their home ports, but would serve a far greater strategic role than merely approaching another fleet and blasting the hell out of it.
Image by James Clyne
Stable Lagrangian points around planetary bodies could essentially be contested 'ground' from which to defend one's home turf with a minimum of effort. Maintaining operations in orbit allows for an incredible amount of defensive work to be carried out. Orbital gun platforms and ships could easily calculate and test the best ways to blow targets to small pieces with an ease of effort, especially if you know where an enemy force is generally coming from. Looking at this from a branch perspective, you'd likely have a specialized group that works extensively for orbital defense, akin to the US Coastguard or Navy, but with more specialized concerns.
Low Earth Orbit operations is generally confined to civilian space agencies, but there was a time when the military did look at space as the highest ground from which to blow each other up. There's an element of this that concerns itself with the ground, such as when you use an Intercontinental Ballistic Missile, but when you have the possibility of someone coming from higher up, a LEO element to supplement orbital operations would likely be used, either for interception (think Starfighters) or observational purposes (think spy planes or satellites). Want to place a spy satellite up in orbit? Want to stop someone from placing a satellite in orbit? Or, do you want to take out said satellite after you find it in orbit? These sorts of forces could conceivably be used to deter and harass any sort of invasion force.
Finally, as Hitchhiker's Guide the Galaxy notes, space is really, really big, and placing an armed force on the ground would require the ability to keep said forces alive between the time it takes to get from point A to point B. A non-strategic space force would be needed for a multi-purpose role: transporting, protecting, landing and supporting your hard surfaces forces, all while holding off anyone that your enemy has put into orbit to prevent this sort of thing.
This is the branch where your popular space marines would likely fall into. On the very small possibility of ship to ship boarding, you'd want a contingent of soldiers who would be specially trained for combat in space, boarding actions, sabotage or a liaison between ground and space-based forces.
Putting a military into space is really complicated, and it requires a lot of information in order to function. Any space-based military would likely be data driven. Strategically speaking, a nation's military commanders would need to have a good idea as to what any opposing force would be up to. In addition to conventional intelligence (ie, someone on the ground who's watching and reporting), that involves a lot of calculation into just how an orbital force would be moving around. On the other side of the lines, their counterparts would be doing everything possible to prevent that sort of accurate information from getting out.
Military history is rife with points where misdirection is used to give an enemy force the wrong impression of what they're capable of: look no further than the inflatable tanks in the North African Campaign or faked deployments in England prior to Operation Overlord during the Second World War.
On a tactical level, the ability to cripple one's assets (and prevent tampering) onboard a spacecraft would be an essential task for anyone in orbit, in addition to calculating information and jamming enemy weapons and sensors. Electronics warfare is in and of itself a new, real world field that is largely uncharted territory, but one that's critically important when it comes to managing one's infrastructure. This would extend right down to the ground, which makes me think that you'd have an independent branch, because electronic warfare could extend into a wide range of arenas.
All of these various branches are contingent on one thing: a central, military objective that exists for a governmental military. A rock thrown from one side of the solar system to another is one way of starting and stopping a conflict, but the same has been said for the introduction of the atomic bomb on the battlefield. These are the ultimate strategic weapons because they have the potential to eliminate life on a planetary level, and if such actions aren't explicitly banned by various parties, their use would be extremely limited, and likely only used in 'testing' scenarios or displays, or by fundamentalist groups that have absolutely nothing to lose.
When people go above the skies, they're going to take all of the complicated and messy parts with them: this includes our propensity to get into fights and play politics for one's own group of people, and given enough time and resources, warfare will inevitably break out in some limited fashion. Why go to war? There must be some achievable, stated objective that requires and appropriate level of force, beginning with a strategic level of conflict (ie, no shooting, but with the presence of people who have a high probability of causing a lot of problems), down to a tactical level that seeks to undermine an enemy's strategic disposition, or maintaining some level of force to quell troublemakers who aren't aligned with any particular political body.
The end lesson for waging war in space is oddly an optimistic one. For all of the many ways that holding the ultimate high ground in space can be used to eliminate people in an absolute fashion, the necessity for maintaining diplomatic cooperation between hostile political factions is the most expedient means for keeping everyone alive, provided that politically, everyone is a somewhat rational player. Even if they're not, the sheer logistical scale that goes into fielding an armed force across the vast regions of space is nothing to sneeze at.
If you do put said force into play, it’s clear that there’s going to be more on the front lines than regular Earth-based analogs of the Army, Navy and Air Forces: there’s going to be a whole new generation of branches that will be put to the test in the depths of space, each one specially suited for their line of work.