Stealth cards cannot be directly targeted by your enemy. This means they can't target it for attack, or cast a spell on it, or use a targeted effect on it.
However, it is NOT immune to untargeted effects. There are three types of untargeted effects:
- Full-Board: Cards like Arcane Explosion ("Deal 1 damage to all enemy minions.") that affect an entire side of the board.
- Area of Effect: Cards like Betrayal ("Force an enemy minion to deal its damage to the minions next to it.") that affect minions based on position.
- Random: Cards like Knife Juggler ("After you summon a minion, deal 1 damage to a random enemy.") that affect a random minion.
Stealth does not affect you; that is, you are able to target your own stealthed minions at will.
Similar to Stealth is Untargetable (for lack of an official term), an effect shared by Faerie Dragon and Laughing Sister. Untargetable minions can still be hit by untargeted effects. Unlike Stealth minions, however, they can be targeted by minion effects (Elven Archer's damage, Spellbreaker's silence, etc.), but not by the player that owns them.
As for your claim that some minions were not affected by Arcane Explosion — is it possible that they had Divine Shield or were under the effect of Commanding Shout?