DC’s:
One side starts at ten because the other side gets a d20 roll and the expected value of a d20 roll is, er, 10.5 which turns out to be close enough.
Each side then adds their bonuses from there. The DC is the side that starts with the ten for opposed spells and abilities. When you cast a spell that’s you.
There are a few abilities like Parry in which both sides get a roll but in those cases neither side gets the ten to begin.
This is actually kind of impressive in that it's accurate, but hard to follow even for someone who knows how the rule works.
DC is the difficulty check for a save or skill. It's the number that has to be met in order to succeed. In most cases, a DC will start at 10 and then be modified accordingly. So in the case of a spell you're casting, a DC is 10 + the level of the spell + the casting ability modifier + any other modifiers for feats, gear, etc.
So for instance, if you're a wizard with 16 intelligence and spell focus (illusion) casting colour spray the DC would be 10 (base) + 1 (1st level spell) + 3 (16 int) + 2 (spell focus) = 16. This means that the enemy you're targeting has to roll 16 or better on the saving throw in order to successfully save against the spell.
The saving throw against colour spray is Will. So if the will saving throw modifier is +3 (say, +1 for the base class saving throw and +2 for a 14 wisdom), the enemy needs to roll 13 or higher in order to save (13 + 3 = 16).
Since a Will save negates colour spray, if the enemy rolls 13 or better, they will not be affected by colour spray in any way. This gives the colour spray a 60% chance ((13-1)/20) of hitting the enemy with its disabling effects. Naturally, this means that if the enemy had a high will save, such as +9, we wouldn't want to use colour spray since our DC of 16 means the enemy would save on a 7 or better (9 + 7 = 16) which means it has a 30% chance of success. We'd be better off targeting a different saving throw that's lower (can be seen using the inspect creature interface) or maybe not targeting saves at all, depending on the circumstance.
Skill checks work similarly to the saving throw situation described above, except normally the skill check DC is static and set according to the difficulty of the action. So say, climbing a tree might have a DC of 15 and we'd check athletics against it. This skill check will be modified by skill, the attribute (strength for athletics), and the armour check penalty. This could be modified by other elements such as gear, buffs, and so on.
Finally, as
Desiderius mentioned, there can be times where a skill check is contested, which means that two creatures are trying to do something and both roll a d20 + modifiers and the one that has the higher number wins. This doesn't come up nearly as often but as long as you get the gist of what's going on, you should be ok. Generally speaking: higher modifiers for you = good, higher modifiers for the enemy = bad and the one that you most have to pay attention to is the spell DC stuff described at first. Otherwise, you can just know that higher = better.
tl;dr There are a lot of rules since these games have a lot of tabletop history and most of it was adapted into the game. It's difficult for the developer to actually explain everything for someone who doesn't have some passing knowledge of the rules history, but the game is quite good if you can get past the learning curve, and some of the nuances are more overwhelming than complicated.
Unfortunately, 3rd edition D&D which Pathfinder is based is the most rules heavy edition of D&D and Pathfinder 1st edition made it somewhat more complex, added a TON of classes, and also made some subtle changes so that if you think of it only as 3E D&D, you're going to make some weird mistakes. I find the number of classes in the Pathfinder games to be a little overwhelming and my first contact with D&D rules started with the basic red box set, then 1st and 2nd edition AD&D somewhere around 35 years ago or so.
