The three d20 rolls
Almost every "does it work?" question in D&D is a d20 plus a modifier against a target number:
- Attack rolls against the target's Armor Class.
- Saving throws against a spell or effect's save DC.
- Ability checks, skills included, against a DC the Dungeon Master sets. Initiative is a Dexterity check.
Natural 20 and natural 1
On an attack roll, a natural 20 hits whatever the target's AC, and it is a critical hit: roll all the damage dice twice and add them together. A natural 1 misses whatever your bonus.
The 2014 rules give no automatic success or failure on ability checks or saving throws, though plenty of tables play it that way. Say which rule you use before the session, not after the roll.
Death saving throws
At 0 hit points you roll a d20 at the start of each turn. 10 or higher is a success, lower is a failure. Three successes and you stabilise; three failures and the character dies. A natural 20 brings you back with 1 hit point. A natural 1 counts as two failures.
Rolling with advantage
Two d20s, keep the higher. It raises your chance of a natural 20 from 5% to 9.75%. It has its own roller.