From D&D 3.5 to 5.0(5E) - 7 - Advantage and Disadvantage in 5th Edition Rules

One major change in combat rules, and in other situations like saving throws and skill checks, is the idea of "Advantage".  Advantage is the mechanic in 5.0 to show that the character should do better than average in a situation and disadvantage is when the character should do worse than average.  Both advantage and disadvantage are handled by rolling two d20s when a character would normally roll one.  If the character has advantage, the character will take the more favorable roll.  If the character has disadvantage, the character must take the less favorable roll.

Dwarves, for example, have an advantage on their saving throws versus poison.   So when everyone in the party is making their saves versus the poison gas, the dwarves in the party each get to roll two d20s and take the higher roll for their saves.  Characters wearing armor that have a stealth disadvantage have a disadvantage on dexterity checks.  That means they will be rolling the d20 twice and taking the lower of the two rolls.
You may have noticed that a lot of the rule changes in 5.0 eliminate a lot of math.  We do a lot less adding one for this, subtracting two for that, adding another two for something else.  This elimination of many minor adjustments should result in speedier play.
A few more examples of advantage and disadvantage:

• 7th level barbarians gain advantage on their initiative rolls. 
• If you are within 5 feet of someone and you want to shoot them with your crossbow you have a disadvantage, and must roll 2 attack roles and take the lower roll.  Unless you have the Crossbow Expert feat, then that disadvantage no longer applies to you.
• If your enemy cannot see you, then you have an advantage on your attack rolls against it.
• If your enemy is the recipient of a blur spell, then your attacks have disadvantage against that enemy.
• If you are the subject of an Enlarge spell, you gain advantage on Strength checks
• If you are the subject of a Haste spell, you gain advantage on Dexterity saves

Advantage and disadvantage is used extensively in 5th edition rules; so it is one of those core rule changes that you really need to understand.
Read more about it on page 173 of the PH, but you will find cases of it all over throughout all the books.