For almost as long as the human race has existed, they've been perfecting the art of poking holes in other humans to kill them. Coincidentally, a large amount of time and effort has gone into thwarting the hole pokers. The race between arms and armor has been a never ending, constantly evolving struggle that will likely continue until we call it quits as a species and bequeath the planet to the robots.
I could go on about the various types of armor throughout history for days. Everything from stiff leather to metal suits have been used. It fell out of fashion once firearms that could reliably pierce even the heaviest plate came about, but once people figured out that layers of sturdy, densely woven cloth could stop bullets reliably, the race was on again.
To learn how to defeat armor, one must first understand how it works. All armor, from the simplest boiled leather to the most modern beskar'gam, relies on one simple principle. The vast majority of weapons, from knives to swords to guns, function by focusing as much kinetic energy on as small an area as possible. On a blade, that energy is focused on the edge. A sharp blade can cut through skin, flesh, and bone with ease if there's enough force behind it. A bullet is a small, dense object travelling at high speeds. It strikes with tremendous force, and concentrates that force on an extremely small area.
Therefore, the job of most armor is to take that force, whether from bullet, blade, or arrow, and spread it out over a wide enough area that the armor can keep it from penetrating through to the flesh below. Chainmail uses interlocking rings to spread the impact of a blade out, while scale and plate armor use shaped metal sheets to accomplish much the same. Bulletproof vests rely on tightly interwoven fibers that are, pound for pound, stronger than steel to arrest the forward momentum of the bullet and spread its energy out over a more manageable area. Steel strike plates take the concept further by spreading the impact over a much, much larger area, and ceramic strike plates take the concept of energy one step further by deforming and shattering on impact, soaking up a lot more of the energy in the process.
In Star Wars, there's another element to content with as well: thermal energy. Everything from blasters to lightsabers rely on thermal energy to do the bulk of their damage. With that in mind, Star Wars armor has to be an effective insulator on top of everything else.
There are a couple of schools of thought when it comes to negating the advantages offered by armor.
The first is to defeat the armor itself. This is most easily accomplished by piercing it. A good dagger will bust through the links of most chain mail. Though it's good at stopping slashing damage, it's not so good at stopping stabs. Most modern cloth armor like Kevlar is similarly ineffective against a sharp point. It might seem counter-intuitive to think that a vest that can stop a .45 bullet can be beaten by a $10 pocketknife, but that's the truth.
Plate armor is a little more difficult to pierce. But, as the saying goes, God made all men, but Samual Colt made all men equal. Thin sheet metal is surprisingly ineffective against bullets. You know the scene in the action movies where the cops all take cover behind car doors and the like? Yeah, the thing is, even a relatively anemic round like the .38 Special will go through a car the long way, so long as you don't hit something solid like the engine block. You can increase penetration power by using special Armor Piercing rounds, which are basically just normal bullets with a solid, dense core. Or if you're really feeling froggy, bust out the armor piercing incendiary rounds. A good API round will cut through thick steel like a knife through butter. And if that won't do the trick, shaped charges work wonders.
And if that's not enough, you've got a plethora of energy weapons that can cut through most armor like so much rice paper.
But what about the exotic crap that can't be cut with a lightsaber, much less a bullet?
Well, that leads us to the second way to beat armor: if you can't defeat it structurally, defeat the flesh beneath.
Recall that the purpose of armor is to spread the energy of an attack out. With few exceptions, the armor itself is supported by the body. If you can't punch through, you can still make it smack the wearer hard enough that it really doesn't matter.
For a real world example, look at bulletproof vests. The Army's vests will certainly stop a bullet, but that kinetic energy has to go somewhere. Usually, that somewhere is the rib cage. You'll most likely survive a shot to the chest wearing one of their vests, but it's going to bruise the everloving hell out of you, probably crack ribs, and potentially even cause internal bleeding.
Beskar can stop just about anything short of a turbolaser strike, but you hit that bitch square on with a .50 BMG round, and you've just transferred the energy of a car crash to an area the size of a dinner plate. If your opponent shrugs that off, you've got ample reason to be a wee bit incredulous. But what if you're not carrying an anti-materiel rifle around?
In that case, take a leaf out of the Imperial Remnant's book. In Invincible, they faced down a strike team of Mandalorians in full beskar'gam. Their blasters couldn't penetrate. So what did they do? They just concentrated their fire and cooked the bucketheads in their armor.
And let's not even get into the subject of sonic rifles. Or concussion grenades. Or any one of a hundred different other things that might not break the armor, but'll break the person wearing it.
People have this idea that armor makes them invincible. It does not. Use your head, use your weapons, and fight dirty. As always, if you have any questions or comments, the comments section and my inbox are both open.