Now for the longer post . . .
What specifically a set of BA does is dependent on two things- movement profile and weight range.
Movement profile
Does it jump and does it get 3+ hexes of movement as a option?
Yes
Most useful IMO is the suits that can jump for 3 or more hexes which is how we were introduced to the original BA and still gold standard- the Clan Elemental. The Elemental suit meets several design thresholds- can take a 10 point hit, generate a +3 to hit (+1 for BA tactics, +1 for jumping, and +1 for 3 hexes or more), and has a high burst damage value especially with later weapons like ERMicro & APG. The IS tried to duplicate it, which resulted in the IS Std and faction specific Raiden, Cavalier, and Longinus- the later is the closest but b/c its built by the IS is unable to jump before it clears its missile packs though none can take the same damage. Later with Recoilless Rifles, BA gauss rifles, Plasma Rifles, and MagShots the IS designs gain some range over the original trio of weapon choices- SL, MG and Flamer. I will explain below why jumping for 3 or even as some suits do 4, is important in a combined arms strategy.
No
Some of my favorite BA for campaign/play purposes do not jump but have 3 MP, it just limits their ability to be deployed and involved in attacks on mechs. Trinity (Asterion, Theseus, whatever the Cappie name is), GDL Standard, and a few others get that +1 TH from their movement which helps with their longevity and can be used as a trooper but unfortunately they are just not as good over all. Typically though for not getting 3 hexes or more of movement you are talking about area via heavy weapons in the suit designs- though some of them jump. A lot of the suits the IS came up with in the late 50s and 60s to improve the throw weight of their BA fit in this category- heavy & assault sizes that carry lots of SRMs or guns to create a Hunchback style bubble where a enemy player does not wish to go if they do not want to take damage. Consider the Grenadier a lighter and easier to place SRM Carrier. The Kanazuchi is a ML turret in its original version. Use your transport assets to place them in key terrain on the map and you can control the shape of the battlefield.
Deployment (Why jumping or not being assault size is good)-
No matter what, BA are generally slower than mechs and even a lot of armor options. They are still faster than most infantry, but it does not help when you want to use them against tanks and mechs. Enter the mobility options . . . Omni-Mechs and vehicles can load BA without suffering any penalties though it does take a MP to load and a MP to unload which must be done at specific times and IIRC the BA must be done with their movement- to be honest, I do more maneuver warfare with MM than table top. Table top I usually have to just unload from pre-loaded carriers- only a few times have I had enough table time to move a transport to pick up and re-deploy a BA squad somewhere else on the table. Assault BA or suits lacking the proper manipulators will not be able to mount Omnis.
Certain BA however CAN load on standard mechs or vehicles- the BA themselves must mount the equipment called 'Mag Clamps' to be let them hold on. It does slow down the mech but its often still faster than having them move 2 or 3 hexes across the map- especially if its attached to the right vehicle. The Harasser hovertank moves 10/15 . . . they could redeploy a Marauder BA squad at the speed of 9/14 after taking on the suits, quickly allowing a player to adjust to changing conditions.
Besides the +1 TH, jumping also allows any BA loaded on a VTOL or other aerial craft to deploy the BA without having to land. Landing makes the VTOL vulnerable (loses its VTOL TH bonus) as well as typically absorbing multiple (3 or 4 MP) movement points- or even risking collision with terrain. Ideally your VTOL flies over a heavy wooded hex that your Grenadiers jump out to occupy- having to land, like with a Ogre squad, means it take 2 turns for the non-jumpers to occupy the terrain and makes them extremely vulnerable when landed. Its a comparison of +4 TH (+2 heavy woods, +1 BA tactics, +1 jumping) for the Grenadiers vs the Ogres getting a +1 TH (BA tactics) sitting on open terrain where the VTOL landed. If its a larger VTOL like a Shun or Karnov BA, there is a good chance it also places OTHER assets (BA or infantry) at risk if someone shoots at the VTOL. This contributes to making jumping being one of the most important design factors in BA.
Tactical uses
Pursuit & Screening-
In my experience, often with larger play areas and bigger (company+) battles, BA execute several different missions very well. To me, as a Clan BA player, pursuit & screening are the most important role though its part of a aggressive BA play style. One example I often use was a battle of a medium nova (5 mechs & 5 Elemental points) against a heavy or perhaps assault IS company set in early 3060s BV balanced. My slowest mech was a Timber Wolf E as part of a typical Wolf style force- probably had a Ice Ferret and pair of Adders with another Clan med. We started off out of range so everyone was running to close, especially as my Wolf force out-ranged the IS company. Once I started to be in range, but before he had too much in range, I dropped the Elementals off the Omni-Mechs behind a Level 1 difference in a area that had several 'holes' in the hill where the LOS would be blocked under MM's double blind rules. The Wolf mechs started walking in reverse to keep the range open- to close the range on my mechs he was going to have to pass where I dropped off the Elementals. He knew where the Elementals dismounted, but it was taking his 3/5 and 4/6 mechs a few turns to get to the Elemental line so they were able to reposition. Several points jumped out to flush their SRM racks, building on the damage my lasers and missiles had already caused but they also caused that player to try to decide what he shot at and how he moved to avoid the Elementals- which helped me keep the range open and his weapons out of play. He moved his pristine Banshee (7S? 5S?) into one of those 'holes' in the hills to benefit from the partial terrain. A Elemental point leaped afterwards into the same hex, surprising him . . . The Elementals had not fired a shot and were the last point to be revealed. My Elementals successfully made a leg attack and for the crit roll got a 12- leg blown off. The pristine mech dropped into that pot hole and was unable to effect the rest of the battle- and with the damage the rest of his company had suffered started a general retreat. The terrain had that single elevation hill, several of those pot holes and a few light woods . . . it had slowed the IS company's chase of my Clan mechs, but now when they had to retreat it slowed them even more- only being able to walk backwards on mechs that have a walk of 3, 4 or 5 MP means they cannot avoid or pass through terrain very easily. My Elementals kept jumping forward in pursuit, firing SRMs while they lasted and their SL/MG/Flamers when in range and trying to get more leg attacks on the slower assaults. No more legs blown off, but I did get a few minor leg crits from the Elementals before the battle was over. The point here is that Battle Armor are very good for creating a obstacle against a force that is trying to rush your forces. I have been on the other end, trying to get around a Lyran player's Fenrir BA to get to send my flankers in at his fire support mechs- but those Fenrir run 4 hexes and come with ERSL, SRMs and MLs to be mobile bubbles of pain.
Against players who field slower machines on anything but a Kansas wheatfield, 3/5 & 4/6 speeds, your 3+ jumping BA have a chance to keep up, once they are in range, and contribute to your mechs/tanks attacks. IF a mech gets knocked down, they can close on it to keep up the hurt since it will spend at least 2 MP getting up. To avoid your BA an opposing player might have to run which hurts their targeting numbers if they have to turn, pass through woods/rough, or climb any elevation- now that 3/5 that ran only moved 4 hexes so they get a +2 for their firing numbers while your mechs/armor only have to deal with a +1 TH. Sometimes that is enough- the threat of what your BA can do to them if they do not move to avoid them- jumping lets you ignore the terrain they have to move through at cost. This just gets better when you start talking about BA that can jump 4 hexes like the Void and Clan Med BA, now you are a definite threat to the 3/5s and start to threaten 5/8s.
More tactical uses would be spotting, seizing objectives, urban warfare to be covered in later posts (out of time atm)
What not to do-
Take a single BA squad/point to determine how they are used
Some folks seek to try out BA by bringing a single squad/point . . . and then complain when its wiped out for very little gain. Usually it means they positioned it wrong- unloaded in the open within 9 hexes of 2 or more heavy/assault mechs and probably too early as an example. Or it had no transport so that Grenadier was marching across the board 2 hexes at a time. For battle armor to be successful you want to deploy them en mass- the obstacle example above had five points of Elementals which let me create a line 15 or so hexes long across the map on the most direct route for his company. To learn BA, I think you need to bring 3 squads/points complimentary suits- or even the same suit so you do not have stat complications.
Swarming is a Bad Idea
Swarming mechs is a really flashy in the books, but it is a attack that takes 2 turns for effect and can be stopped before you get to attack. When I am facing BA, I WANT them to try to swarm my mechs . . . because that means they are not shooting at me or attacking my legs. I have drowned BA in depth 1 and 2 water hexes that were trying to swarm . . . or taken them into a fire to cook. Besides the typical stop,drop & roll defense.
Unload them in a flat open area
Mentioned above, it is usually one of the classic blunders. Your BA are at their most vulnerable when they unload from transport. Very few suits are designed to deal with it- jumpers can get a +1 if they leave a VTOL, but otherwise they are all restricted to the +1 BA tactics. This is especially bad if they are in range of infantry. Ideally you want them behind a 1 level terrain difference when unloaded, be it a hill or down in a sink hole. The exception to this is the Purifier and the few other mimetic armor BA- they get bigger defensive bonuses if they do not move, and unloading does not count. Like most rules, there are times to break the rule but it takes a experienced BA user to know when and what payoffs to decide on. Dropping a single unit in a area is also not as good as dropping multiple units. Now if someone gets the chance to fire (won initiative and have last move) you at least force a choice and they may try to split their fire rather than eliminate a single squad/point- this also helps protect the transporting unit. I like it when someone tries to shoot at my Ice Ferret that dropped off the Elementals, the Ice Ferret has a +3 or +4 TH for their movement while the Elementals will just have the +1 for BA Tactics. Split fire at the targets? Now you added a +1 TH for the secondary targets . . . and anything (mech, veh or BA) with Stealth Armor cannot be secondary targets.
Pay attention to the opponent's weapon selection, fire is bad
For most BA you want to stay away from area affect or heat damage- this means artillery like Arrow IV or snub nosed pieces like the Sniper or Long Tom Cannons and such things as Plasma weapons & inferno SRMs. Artillery does their damage to each individual suit rather than single hits. Hitting a Elemental point with a AC/10 means a single trooper takes 10 pts of damage. Having a Sniper round land next to your Elemental point means EACH trooper takes 10 points (or whatever splash is) which makes them nearly dead since they take 11 points of damage. It will kill medium IS suits and severely damage the heavier ones. Plasma Rifles and Cannons can hit BA at ranges most cannot respond, and they will do variable damage (roll a set number of dice to get the total) to the whole squad/point. While it may not wipe out the whole squad/point, it will push most into critical condition. Infernoes will kill suits as well- I think its for every 3 inferno missiles that hit, 1 suit dies . . . one time an opponent dropped off his Fa Shih squad (just the +1 TH for BA tactics) which let me place my Harasser Missile Platform 3 hexes away (3 Gun, +2 move, +1 BA tactics) which let both SRM6 racks hit, for a total of 12 inferno missiles. The whole squad was wiped out without ever doing a thing. Next game, he killed the Harasser fast, lol.
