Since the 3rd edition (or what we now call Pathfinder) of D&D, the mathematical process of the world's most popular roleplaying game has been the same. You roll a d20 to establish a base level of success, and then you boost that chance of success by adding points you've earned during character creation and by leveling up. Your character's innate attributes help some (a +4 Strength modifier, for instance, is helpful when you're prying a locked door off its hinges) but the big boosts come from proficiency in specific trained skills (Athletics, Acrobatics, Sleight of hand, persuasion, and so on.)
The physical tool we use for this is a twenty-sided die (or d20 in gaming terminology.) A d20 represents a percentage of success. Complete success is 100% (that's a 20 on the die). 100 divided by 20 sides is 5, so when you roll a d20, you use increments of 5% when expressing your chance of success. So a +2 to your roll is a 10% (2×5) increase to your chance of success, while a +4 is a 20% (4×5) boost, a +6 is 30% (6×5), and so on.
In practise, though, most actions don't require a full 100% success to achieve your goal. For instance, if you're trying to read ancient runes on an obelisk, you probably don't need to comprehend every word to get the idea of what it's saying, so a roll of 75% might be good enough. A 75% translates to whatever 75÷5 is (it's 15), but if you have a +4 to your roll then you actually only need to roll an 11 (because 11+4 is 15).
The important thing to understand is that your character's bonuses increase your chance of hitting the required number.
Interestingly, although both 5e and Pathfinder both use this exact same system, they manage to express these boosts differently.
In 5e, you choose your skill proficiencies once, during character creation. A proficiency bonus gives you a substantial boost when you attempt an action that falls into a general skill category. When you jump out of the way of a swinging pendulum trap as a 1st level character, for instance, you might have a +2 Dexterity and a +2 Proficiency bonus to your d20 roll. That's +4 to whatever you roll on the die.
As you level up in 5e, you often gain new powers and tricks you can do, but the points assigned to your skills don't often increase. Every four levels, you have the option to boost the attribute that a skill group is based on. It's a weighted system, though, so it actually takes eight levels to get an additional +1 in any single attribute. If you're a 1st level barbarian with a +2 to your Strength, you can add half a point (effectively) at level 4 and another half at level 8 to result in a +3 Strength. There are only 20 levels in 5e, so by level 16 you've only been able to see a 2 point increase.
Not coincidentally, your proficiency bonus increases by 1 every four levels. By Tier 4 (level 15 to 20), your proficiency is +6, or a 30% boost to any roll on a skill or saving throw you have proficiency in.
So in 5e you do increase in raw power, but gradually over time and broadly across yourproficiencies. That transates to armour classes (AC) and difficulty classes (DC) that stay mostly within the range of 5 to 20, with 25 being an occasional upper limit. It also means that the leveling up process is relatively low-maintenance. When you increase in level, you reference your class to find out what new abilities you might have earned, you roll to increase your hit points, and that's it. Only every four levels do you have to increment your proficiency bonus or increase an attribute score.
It's elegant math, and I don't know of anyone with a mind for systems design who has played 5e who doesn't admire the way 5e keeps its numbers simple and manageable.
In Pathfinder, you choose your skills once during character creation, using a point buy system called ranking. You're given a budget of points, which you can assign arbitrarily to any number of skills you want. When you put a rank into a class skill, however, you get a special instant bonus on the assumption that your character class would be especially proficient at certain things.
It's hard to accidentally build a rogue, for instance, that isn't very good at stealth. Stealth is a class skill for rogues, which means you get an automatic +3 bonus to stelath. That starts at level 1, and it potentially increases as you level up.
The class skill mechanic is the equivalent of the proficiency bonus in 5e, and in fact in Pathfinder 2e is also called proficiency.
When you level up in Pathfinder, you gain some number of additional points to spend as skill ranks. You can choose to specialise in a few key skills, or you can choose to spread your skill ranks around and become a generalist. In fact, you're not allowed to have more ranks in a skill than your character level, which is how the system balances skill proficiency and keeps it at a predictable rate of growth. The exceptions are the class skills, with their initial +3 bonus, but that's the point of a proficiency.
Because you have control over which skills you improve as you level up, there's the opportunity for you to focus on skills that interest you. And with over 30 skills to choose from, you have lots of flexibility. If you want to build a fighter who also happens to have read every book under the sun, you can rigourously invest in all the knowledge skills as well as the obvious athletics and acrobatics. If you want to build a sailor who can't read, but who's travelled enough to have local knowledge about every place possible, then you can invest in Knowledge (Local) as well as Swim and related skills. In Pathfinder, the mechanics of your character build are a key way of expressing your character's story.
The ramification of increasing skill ranks is that the difficulty classes (DC) must also grow. By level 10, a character could have 10 ranks in a skill for a +10 bonus (that's a 50% boost to success) to a skill check. By level 20, a character could have 20 ranks, which is a +20 bonus (or 100% boost to success) to a skill chance. What gets counted as a success cannot remain at 20 or 25 maximum for the whole game, or else by level 20 no character could ever fail at anything. For that reason, the DCs in Pathfinder get higher as the game goes on. It's not uncommon to see DC 35 or DC 45 in high level Pathfinder adventures. A level 20 character with a +20 bonus still has to roll a 15 or the die (or just a 12, accounting for a class skill) for success. In other words, the definition of success is a sliding scale in Pathfinder.
The process of leveling up is a little more rigourous in Pathfinder than in 5e, because not only do you have to add new class abilities to your character sheet, roll your new hit die for more hit points, but you must also buy skill ranks. For the little bit of extra work, though, you get a flexible and robust system that ensures that you can build exactly the character you have in mind, and that your character concept is reflected, as accurately as possible, through the game mechanics.
The bottom line:
There's an easy and colloquial way to sum all of this up, and that's penalties. In Pathfinder, penalties tend to come in varieties of -2, -4, -6, and so on. You roll your dice, you add your modifiers, and then you subtract your penalties.
In 5e, a penalty is often expressed as disadvantage, in which you roll your die twice and take the lowest of the two rolls.
This embodies a lot about each system. Some players love 5e's disadvantage system because it involves no math, it's quick, and it produces a result, even though it does effectively reduce the problem to a 50-50 chance. Other players love Pathfinder's system, because it lets the player retain every last percentage of the chance for success, even though it does involve more calculation.
It's OK to like them both, of course, and it's fun to understand the difference.