Alright, Dungeons and Dragons in one paragraph?
DnD runs off the D20 system, a die with 20 faces. It literally runs off this. Virtually everything in the game relies on two things.
1: You roll a d20, which determines if you succeed or fail. Default 'difficulty' is 10, so you need to roll 10 or above to succeed, so you get a 50/50 chance. Difficulty of a roll it determined by the check you're making, like walking up a hill (5) is easier than climbing a rope (15). The roll is also modified by your skill and the enemy's skill, if applicable. So if you're good at climbing, then you get a +3 to a roll, so you get 1d20+3, making it more likely for you to succeed. If the enemy is involved, then their skill is factored against yours, like if the rope is greased then the difficulty would go up, maybe from 15 to 20.
2: You roll for any value involved, if needed. Mainly for attacks, to determine how much damage is done.
Example: You attack an armed goblin. You have a strength bonus of +2, and a special skill that gives +1, while the enemy has a default armor of 10 (that's your 50/50 chance), wears armor that gives +1, and has enough dexterity to dodge with +1. That means the 'difficulty' of the attack is 12, and you have 1d20+3, so you can roll the dice from 4 to 23 and need to hit at least a 12, so you've still only got about 60% chance of landing a hit. IF you win the roll, then you roll for your attack, which depends on your weapon. If your sword gives 1d4 damage and your strength is still +2, then you do 1d4+2 damage, 3-6 damage dealt. Then the goblin would attack you, depending on the goblin's strength, and your dexterity and armor.
This extends to pretty much anything. Climbing - difficulty of the climb compared against your +climb and +dex. Bluff - difficulty of the lie you're making compared against your +bluff and +charisma. Figure out what you're looking at - difficulty of how strange the item is compared against your +knowledge and +intelligence. ('Knowledge (Dungeoneering)' is a skill, as are 'Knowledge (Magic)' and other types of knowledge.)
Most difficulties are set by default. Armor is a static value. If you do something strange, like you decide to steal a boat, then the DM has to make up estimate numbers on how difficult it would be, so the DM's job is to set up issues, move the NPC around, and to make fair guesses as to how difficult or how likely certain things are.
THAT. Is the core of DnD. The rest, equipment, movement speed, damage values, magic types, etc... That's all details that work into this core framework. The core is still right there. Any given item or stat will change the difficulty or the bonus you get.