I have never played D&D, but I would very much like to play one day, when I'm not as busy, plus when I'll have a group to play with. (That's important I guess).

My question for today is: when another player gets attacked by a monster or something, obviously it's not my turn: can I interfere in some way?

For example if I am an archer, can I shoot an arrow at the enemy to distract him? Or another example: can I jump in front of the other player, and take the hit myself, because I have more hit points, and could handle it?

I'm guessing, no, just cause it's not my turn and thus can't attack, but perhaps stepping in front of the other player and taking the damage. I'll go ahead and say no, but I hope to hear otherwise, I think it would make the game more realistic. If not, that's fine, I would still very much like to play, as it's seems like an awesome game as it is.

No and yes.

The short story is that you can't act out of turn unless you have a specific feature or rule that allows you to do so. You do have a few options for what you want to do, however:

• 
  The Protection Fighting style (available to a few classes, namely Fighters and Paladins at level 2) allows you, if holding a shield, to use your reaction to give disadvantage on the attacker of an adjacent ally. In other words, once per round, you can hinder a foe from hurting an ally around you.


• 
  Opportunity Attacks, if an enemy leaves your melee attack range, you can use your reaction to attack them. This attack can be used to limit their movement and prevent them from being able to get to the ally. This however, does not work with ranged attacks (unless you take a specific feat I can't find right now) or without putting yourself in harm's way in the first place.


• A few features like the Battle Master Fighter's Maneuvering Strike maneuver allow another ally to move up to half their speed out-of-turn. This can be used with an Opportunity attack to let the ally move out of range of the enemy.


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  Readying an action: this is actually the closest to what you want to do. It requires that you use your Action on your turn (say, shooting the incoming enemy) to declare that you will use your reaction to take a certain action when a specific event occurs (say, an enemy approaches a specific ally). This will use both your in-turn Action and your out-of-turn Reaction to do this; you will need to check with your GM that the trigger event is possible and that the action will have the effect you want.

There are probably additional features or spells that can be used in such a way (probably costing your Reaction for the round), but I don't remember having heard of them.

On a more philosophical note

D&D is pretty strict in its combat structure and rules. The turn structure means that you cannot act anytime you want and that some things are just not covered at all. For example, throwing yourself in front of an ally to take a blow is not covered at all, and your GM has the right to just say "You push Alice 5 feet to the right, the orc moves 5 feet further and hits." You have to find a way to express what you want from the rules of the game. Or ask the GM to make something up.

Not every game is like that: some give much lighter rules and expect the players (and GM) to add personality to combat by their description, their choice of action and circumstances. Don't be afraid to look at other games if you find a group that uses one. Most bring interesting twists and mechanical flavor.

An amusing example compared to D&D: The Powered-by-the-Apocalypse games take a super narrativist approach to combat, such that there is no initiative, everyone can act when they want and the GM is expected to let every one have a moment to act and shine. Even big enemies can be almost killed in a single hit from a combat-focused (and a bit lucky) individual.

If you didn't plan ahead, then no, you usually can't.

There are abilities that allow you to protect others. Defensive spells that can be cast on an ally (I completely swung a recent combat with a simple +1 to AC from a Warding Bond spell), actions you can take to give a foe Disadvantage on their attacks, etc. If you didn't do any of that, though, there are very few options to help once the foe is already attacking, outside of specific class/archetype combinations that can spend their Reaction to defend someone.

Other game systems sometimes do allow this - I did exactly this in a Star Wars Edge of the Empire game by spending a Destiny Point to say "I dive into the girl and take the blaster shot intended for her."

Yes, if...

If you build your character around protecting your teammates.

Paladin Oath Of The Crown (Sword Coast Adventurers Guide, p. 132-133)

Divine Allegiance

Starting at 7th level, when a creature within 5ft of you takes damage, you can use your reaction to magically substitute your own health for that of the target creature, causing that creature not to take damage. This damage to you can’t be reduced or prevented in any way.

Any paladin can take the Protection fighting style

When a creature you can see attacks a target other than you that is within 5 feet of you, you can use your Reaction to impose disadvantage on the Attack roll. You must be wielding a Shield.

These are two examples, there are many ways to protect other players. The Oath example seems to closely match your question about taking a hit for a teammate. A similar one is the Aura of the Guardian in the Oath of Redemption Paladin

Aura of the Guardian

Starting at 7th level, you can Shield others from harm at the cost of your own health. When a creature within 10 feet of you takes damage, you can use your Reaction to magically take that damage, instead of that creature taking it. This feature doesn’t transfer any other Effects that might accompany the damage, and this damage can’t be reduced in any way.

At 18th level, the range of this aura increases to 30 feet. (Xanathar's Guide to Everything, p. 39)

For a non Paladin way to help a teammate the College of Lore Bard has Cutting Words

Also at 3rd level, you learn how to use your wit to distract, confuse, and otherwise sap the confidence and competence of others. When a creature that you can see within 60 feet of you makes an attack roll, an ability check, or a damage roll, you can use your reaction to expend one of your uses of Bardic Inspiration, rolling a Bardic Inspiration die and subtracting the number rolled from the creature’s roll. You can choose to use this feature after the creature makes its roll, but before the DM determines whether the attack roll or ability check succeeds or fails, or before the creature deals its damage. The creature is immune if it can’t hear you or if it’s immune to being charmed.
D&D Beyond

Most of the other answers approach this in terms of how to use Reactions to get what you want, but it may be better to try to make sure that it will be your turn first so that you can take actions before your opponent.

For example having the alert feat would prevent you from being surprised, and high initiative could let you act before the enemy archer. On your turn you could try to kill or disable them before they have a chance to attack (Attack or similar action), use an obscuring effect that would make your friend hard to shoot (like the fog cloud spell) , or try to scare off (intimidation check) the archer. This could be narrated as reacting to their attempt to attack, since you did act first.

Initiative only helps you act first at the start of combat, so this wouldn't achieve the narrative you want during the middle of a fight. Note that most fights last less than 3 rounds.