Oh boy, I love military jargon. A tense exchange over a radio in the thick of things is probably one of my favorite things to write. Here's a couple of tidbits I've gleaned over the years that I didn't see mentioned earlier in the thread.
Edit: in retrospect, this has a lot more general military info than jargon in it. I started writing but then I really didn't want to grade these papers so I kept writing and then this happened.
Caveat: I have no actual military experience, just knowledge I've picked up over the past decade from reading and writing.
-- Knowing the hierarchy and chain of command is very important when writing infantry scenes. For example, a fireteam (because you should always have a soldier at your back) is two-four soldiers, and act as a single unit. Squads have multiple fire teams, and farther up there's platoons, battalions, and so on. Each level has it's own CO (commanding officer), which may be an experienced enlisted (no officer training) or a graduate from an officer academy. The dynamics between a hardened NCO sergeant and his fresh out of OCS lieutenant are a great source of tension. Addendum: know what pay grades are and how they're different from ranks. Different military branches will have different ranks in a different order.
-- In a firefight, chatter will be minimal. Rather, it should be: someone who just won't shut up could be a good source of drama. As was mentioned earlier, military communication is designed to be as brief and clear as possible. The fireteam or squad leader (a corporal or other enlisted) will organize the team. Expect to see terms like advance, flank, suppressing fire, assist, fall back, hold position, etc. Individuals will call out enemy movements or their status. If everyone speaks the same language, expect each side to use more code or euphemisms.
-- Infantry love their acronyms. Bases and rally points may have brief unmistakable code names (e.g. Pharaoh, Zeus, Ozone .... bonus points for hidden sentimental/comedic value) or practical abbreviations (Forward Aerial Depot = FAD base). Depending on military branch, soldiers in a fireteam may have nicknames, official designations (Charlie-Niner), or may just be referred to by their role or weapon. Weapons and equipment will usually be referred to by a nickname or official abbreviation, e.g. M20, APC, Mk. II, etc. Depending on the scenario, this requires a bit if research on the writers part. If you're in an original or sci-fi setting, you can be pretty creative here, but try and stick to a uniform and predictable system to maintain suspension of disbelief. Remember: brotherhood is a major theme in militaries, so there's a intentionally lot of vernacular that someone outside of the group would not immediately get to create a sense of inclusivity.
-- Military time: use it.
-- Positioning. The X O'Clock method is solid, but also use the cardinal directions. All distances are in meters. Expect soldiers to be able to tell the time of day from the sun's position and to know which way is north most of the time.
Aerial, Naval, and Vehicular
-- Soldiers who pilot/command ships, planes, vehicles, tanks, etc. can get very attached to their equipment. They may name them, come up with personalities for them, develop histories, and so on. This is a great way to build someone's character development.
-- In the air, there are three directions. Know yaw (left-right), pitch (up-down), and roll (clockwise-counterclockwise). In addition, directions are usually given using a coordinate system based on the cardinal directions, with North being 0, East being 90, etc. There are usually two airmen: a pilot and a navigator. Larger planes, such as bombers, will have additional crew manning weapons and other systems. Communication here is essential: the crew are constantly in contact with their COs or controllers, and require authorization for many actions, such as opening fire, retreating, etc. Targets often have code names or impartial designations. Talk is brief and tight and professional. Phonetic alphabet is used often, as well a lot of jargon. Paint: to lock on to a target. No joy: unsuccessful attack. Weapons free: cleared to engage. Weapons hot: opening fire. Hit the deck: descend quickly. Do a barrel roll: hey, wait a second...
-- At sea, battles are often fought at ranges where the enemy is no more than a speck on the horizon. Individual seamen will have specific jobs to carry out during combat. Don't expect someone to be running all over the ship. More likely, they'll just be sitting there waiting for orders from the bridge. I don't know a lot about naval terminology, but I do know the cook is sacrosanct. There's that.
-- Obviously this is in the domain of sci-fi, but modern day space battles can occur. Regardless, there's very little to no real life precedent for this, so it's pretty much fair game.
-- General space: space is not an ocean. In general, space combat is much more closely related to aerial combat than to naval combat. Military vessels are not spaceships, they are spacecraft. Reusable surface to orbit (STO) craft are shuttles. Stages are sequentially used parts of a shuttle that carry fuel and systems needed at different points in the mission. The space shuttle has three stages, for example: the two solid rocket boosters, the large red booster, and the shuttle itself. Stations are artificial satellites in a stable orbit about a celestial body. You don't use 2D coordinates. Distances are measured in kilometers or AU's and speed in (kilo)meters/second, not knots. Vessels can have classes (e.g. Destroyer) akin to modern naval ships, but they're more likely to have some original designation. Try to use aerial terms (e.g. Bomber, fighter, etc.) when possible. Pitch, yaw, and roll return. In addition, the idea of reference points becomes immensely important. Up/down/left/right/forward/backward are all arbitrary in freefall, and when earth is a curved sphere that takes up a third of the sky, it can be too nonspecific to use. Without a reference point to use, giving directions is impossible. Expect to hear a lot of, "The station airlock is our down RP. Advance upward, and flank forward and backward." Ender's Game is a great novel to read concerning this.
-- Modern space: You probably won't have firefights in raw vacuum, but you probably won't have ship-to-ship battles either. Military operations will be confined to boarding ships/stations, both of which have plenty of reference points to use, or evading surface based attacks, such as missiles. Use the same vernacular as you would in infantry engagements, with the addition of RPs and other "spacey" terms, like vacuum, trajectory, pressure, radiation, reciprocal force (never fire a gun in space without first bracing yourself), horizon, terminus (the line between shadow cast by earth and sunlight), orbit, etc. Cat and mouse fights between two spacecraft orbiting earth would also use terms like Hoffman maneuver (changing orbits), burn (acceleration), G's, delta-V (total acceleration needed to adjust to new trajectory in m/s/s), angular momentum, radial velocity, and other things you learned in physics class.
-- Sci-Fi Space: When you have fleets of spacecraft engaging, things get messy fast. The easiest way is to just say fuck it and use naval terminology. This is where you get things like two walls of spacecraft shooting at each other like regiments in the revolutionary war, which is completely ridiculous. If you're anywhere in space, you're moving, FAST. Actual fleet engagements would take place over thousands or hundreds of thousands of kilometers as they passed each other on intersecting trajectories. You'd spend days or weeks waiting, then push a button, then wait a couple of minutes to see if either of you die. More than likely, battles will take place in an orbit over a planet or moon. The disadvantage goes to the party in the lower orbit. This is called being in the planets gravity well: since they're so close to it, they need to expend a lot of energy (a high delta-V) to escape and disengage, which is difficult to do in a battle. This is the space equivalent of having your back against a wall. Also, quick rule of thumb: the lower your orbit, the faster you need to move, so a fleet could use this to outrun the enemy. As for people, it's much the same as modern day naval battles. You'll have a CO of some kind who issues orders that are relayed by the officers and carried out by the crew. The difference is spacecraft will likely be very small, no bigger than our retired shuttles. Expect the crew to be skilled mathematicians, or have a computer that can calculate all of the trajectories and firing solutions for them. Use terms like kinetic bomb (basically a giant bullet fired very fast), magnetic acceleration (rail guns), relativistic bomb (something small accelerated to near light speed; can destroy entire planets), directional scanners (essentially radar that you point at someone, which they can then detect), passive scanners (much weaker than directional, but also undetectable by others), ion drives (very efficient, and very conspicuous, propulsion), as well as the host of spacey terms mentioned above like delta-V. Everything will be very precise, and there will generally be a lot of waiting. There will be no room for error, so everything will be very tightly regimented and protocol.