Hearthstone has many card effects that are common enough that they get their own keyword - a word or two that immediately tells you how the card works. Let's take a look at the Immune keyword.
Immune characters can't be damaged.
Immune is the rarest evergreen keyword in the game, only getting 1 or 2 cards per set if it ever does appear at all. Immune characters are, as the name would suggest, completely immune to all sources of damage. Just like Stealth, they also cannot be targeted by your opponent while the immunity effect is active. Also like Stealth, a minion cannot be Immune and have Taunt at the same time. If a Taunt minion gains Immune somehow, it will lose Taunt until the Immune effect vanishes.
While Immune characters cannot be damaged, they can still be destroyed (or otherwise affected) by effects which don't target, such as Twisting Nether or Mecha'thun's Deathrattle.
Keywords that Work Great with Immune
A Rush minion can be granted Immune to immediately take advantage of its Rush effect and persist on the field. Giving one to a Windfury minion will allow it to survive both attacks, and Lifesteal will allow it to stay alive longer and heal you more.
Playing Against Immune in Hearthstone
Passive sources of Immune such as Violet Illusionist or Mal'Ganis should be dealt with as soon as possible (and the latter is pretty much mandatory as it gives the hero immunity).
Example Hearthstone Immune Cards
Deathspeaker will give a friendly minion Immune until the end of that turn, allowing it to trade more favorably.
Illidari Felblade is a Rush minion that will conditionally gain Immune if you play it from Outcast, allowing you to develop a large amount of tempo by killing a minion and developing board presense.
Time Out! can be used to protect your hero in a pinch and guarantee your survival for another turn.
Mal'Ganis makes your hero Immune, effectively shielding you from effects that deal damage to your hero so you don't take too much damage too quickly (although this does shut off synergy with cards like Diseased Vulture), as well as giving your other Demons a +2/+2 buff.