Down Point system is a system for generating the effect of "Stagger", "Fall", and "Knocked Down" to mobile suits. Using this system, it creates an opportunity for all suits of all cost to duel against the others, making the game not purely dependent on damage output only.
Let's start by explaining the Down Gauge. Every unit has two down gauge bar. For an example, let's take a look at the BD2 info page here. It's listed in " ダウンゲージ ", with the number 0800/1600.
It means that it has 800 points for it's first (green coloured) Down Gauge bar, and when that 800 points are lost by getting hit, the unit will lose it's first Down Gauge bar and "Stagger" will occur. The state only happens for an instant but getting staggered will make the unit stops doing whatever it was doing (boosting, shooting, walking, etc) and the pilot has to re-input the action to continue.
Are you starting to get the image now? Having staggered while attacking or even just moving about might make the enemy pilot to lose their focus, creating an opening for you to deal further damage or perform any action you wish. But that "Stagger" is not all, let's continue to the second Down Gauge bar.
Still looking at the BD2's Down Gauge stat above, let's look at the second number for the second (yellow coloured) Down Gauge bar; 1600. When that 1600 points are lost, the unit will lose all of it's Down Gauge and will enter the state of "Fall" if it's on the ground (which looks like kneeling), and "Knocked Down" (free falling) if it's in the air. Those states, especially the Knocked Down state, creates a huge opening because it lasts for quite a while.
So how do we get the Down Gauge bar back after it getting decreased by enemy attacks? Just don't get hit for a few secs and it'll refill back up. But if you got hit while the gauge's refilling, it will stop and you will need to avoid getting hit again until it starts refilling back up again.
Now to find out how much of a Down Point your weapons do per bullet. First let's take a look at an example on the left side, a semi auto Beam Rifle. You can see the Down Point it has per bullet shot by looking at the "ダウンポイント" near the weapon's image, in this case, 133. This means to bring the BD2 we see above to the "Stagger" state, you'll need to land 7 shots in one go, and more to bring it to the "Fall"/"Knocked Down" state.
Mark what said above, "per bullet". Some weapons fire more than one bullet per shot, easy example being shotguns. And the weapon stat you see is for one each bullet fired.
Of course, not all units have the same Down Gauge and not all weapons have the same Down Points.
There are three Down Gauge variations, or we usually call it "levels". The first level is BD2's gauge there, which is a 800/1600. Second level is 800/1700, and finally the third level is 800/1800. There are also some units with passive modules, features, modes, etc that reduces the Down Point taken from weapons, so their actual Down Gauge is more than just vanilla Down Gauge listed.
Weapons on the other hand, varies by a lot. From just 30 DP per projectile to 1800 DP per projectile. All of the weapons has their own pros and cons, so just like with damage number, DP number is not everything as well. Quoting the famous Red Comet, "It doesn't matter how powerful it is if it doesn't hit."