From D&D 3.5 to 5.0(5E) - 9 - Resting Overnight, and Short Rests Too

How many hit points will you recover overnight in the 5th edition of D&D?  All of them!  Yes, that's right.  Player's will generally love this, but is it realistic?  Ah, who cares, this is fantasy gaming right!  Besides, parties of high level characters almost always find ways to recover all of their hit points at the end of the day anyway.  It is only the low-level parties that seem to go a few days before their hit points are fully restored. 

If a character with 90 hit points ends the day with 1 hit point, she will have 90 hit points again assuming she gets a good full night's rest.  You might want to argue that a character hurt that bad must have a broken bone or deep cut that would take much longer to heal, but we have been ignoring that game mechanic since the inception of D&D anyway.  A broken bone would more likely manifest itself in a penalty on to-hit rolls or in limited movement.  But, like I said, this is fantasy.  If you want that sort of realism in the game there are no reasons you cannot create your own house rules to cover it.
In the 5th edition your full resting period is called a "Long Rest".  You can only have one "Long Rest" per day and once you have completed your "Long Rest" you can regain all your spells and many other features are reset as you might expect.
The 5th edition named this a "Long Rest" to distinguish it from a "Short Rest"; another new feature players will enjoy.  A "Short Rest", page 67, is a period of downtime that lasts at least an hour, and you can have several "Short Rests" each day.  There are two actions characters can take after a "Short Rest" that I will mention here.  One, found on page 31, is called "Arcane Recovery" and allows arcane casters to recover spell slots equal to half your caster level.  The other action allows any character to recover hit points.  You roll for the number of hit points recovered.  You can roll the die you use for gaining hit points when you level up (your hit die), and you can roll it once for each level.  Thus a 3rd level rogue that rolls d8's for hit points could roll 1, 2, or 3d8 to recover that many hit points after a "Short Rest".  Short rests should allow our characters to stay out in the field adventuring longer before heading back to the home tavern.
Page 67 Long Rest and Short Rest

From D&D 3.5 to 5.0(5E) - 8 - Passive Perception in 5th Edition Rules

The 5th Edition of D&D provides some rules for handling passive skill checks.  Passive checks are used for things done routinely or for checks the DM doesn't want the player to know about.  Each character has a score that can be calculated for their passive skill checks.   I think there are just two passive skill checks.  One is a passive Perception check based on your wisdom score, and the other is a passive Investigation check based on your intelligence.  Your character's passive perception is:

• 10
• + your Wisdom modifier
• + your Proficiency bonus if you have the Perception proficiency
• +/- situational modifiers applied by the DM , such as -5 to notice dangers when characters are traveling fast.

The observant feat gives you a +5 bonus on some passive skill checks.
P175 - Basics of Passive Checks P177 - Hiding
P182 - Noticing Threats

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.

From D&D 3.5 to 5.0(5E) - 6 - Death and Dying in 5th Edition Rules

Dying in 5.0 will not be pleasant.  Instead of the slow drift toward -10, there is a stressful set of die rolls waiting to bring the grim reaper in an instant.

One way to die in 5.0 is via Instant Death.  Now this is not much different than in 3.5.  If your character takes an amount of damage that exceeds her current HP + her maximum HP, she is immediately dead.  So if your cleric with a maximum of 8 hit points currently has 1 hit point and she takes 9 or more hit points in a single attack, she is dead.
The more traditional way to die is to fall unconscious first.  If you are damaged to 0 or fewer hit points, but not enough to result in "Instant Death", then you are unconscious.  From then on, when it is your turn, you roll the d20.  If you roll 10 or higher, that is good news for you because you live to roll again on your next turn.  If you manage to roll 10 or higher on three death saving throws (that is what these are called), then you stabilize.  But if you roll below a 10, and you roll below a 10 three times before you roll a 10 or above three times, then you die.
It may sound like you have at least 3 rounds before kicking the bucket, but if you roll a 1, that counts as 2 failed rolls.  So if you have already failed once, and then you fail a second time with a roll of a 1, you're dead.  If you roll a 20 that counts as two positive rolls.
If you keep taking damage from other attacks while you are unconscious you don't need to count the HP of damage, you just count that as one more failed death saving throw.  If someone cures or stabilizes you, then you are safe from the need to continue rolling death saving throws.
Now remember, D&D is your game and you can alter the rules as you see fit.  Such customized rules that don't strictly follow the book are generally called "House Rules".  Your DM may lay down the alternate house rules for a campaign, or your entire group of players may work together to agree to some house rules.  In my group, we rotate DMs over many adventures during our campaigns, so no DM ever establishes house rules that would affect subsequent DMs or affect how players would develop their characters; we work together to agree on our house rules. 
In my group, I might recommend that we use the rules in 5.0, but that we make it 5 failed rolls before death rather than 3. 
Read it for yourself in the PH on page 197.

From D&D 3.5 to 5.0(5E) - 5 - Damage Resistance, not DR in 5th Edition Rules

I know you didn't like fighting those monsters in 3.5 with DR5, DR10, or even worse DR20.  Well you won't have to any more in 5.0.  Damage Reduction is gone!  But it has been replaced by Damage Resistance.  I don't think the 3.5 damage reduction was difficult to manage, but Damage Resistance in 5.0 is even simpler.   When a creature has Damage Resistance to a type of damage, such as "Slashing", then the damage dealt to the creature by slashing weapons is simply halved.  This allows your first level characters to at least do some damage to creatures with Resistance, but if you can dish out 60HP of slashing damage to an Imp be ready to face the fact that it will only take 30HP of damage, not the 55HP it might have taken in 3.5 when the Imp had DR5/silver instead of the 5.0 "Damage Resistance: non-silver"

And what about the skeleton?  Does he have Damage Resistance?  Not in 5.0.  But skeletons do have a "Damage Vulnerability" toward bludgeoning weapons.  Creatures with damage vulnerabilities take double damage when experiencing that form of damage.
Barbarians in 5.0 will enjoy the benefits of Damage Resistance to slashing, piercing, and bludgeoning weapons while raging.  That helps make up for the lack of bonus hit points that they no longer gain while raging in 5.0.  And it eliminates the dilemma that occurs when the barbarians rage is about to end and he doesn't have enough hit points left to keep him alive.