What is Stealth in Hearthstone?
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 Stealth keyword.
Can't be attacked or targeted until it attacks.
Stealth is a mechanic that causes a minion to be "shrouded" in shadows when it's summoned. While in this state, it cannot be targeted by attacks or your opponent's cards, although it can still be targeted by your own. When a minion with Stealth attacks, it will lose Stealth. The ability will still be present in its card text, although it will no longer do anything. It can later regain Stealth through card effects. While a minion is in Stealth, it can still be affected by abilities which do not target or choose one at random.
If a minion has Stealth and Taunt at the same time, the Taunt will not take effect until the minion loses Stealth. The same is true for Immune.
Prior to patch 188.8.131.5266, Stealth minions would lose Stealth if it dealt any damage (regardless of whether or not it happened via attack). As of this patch however, this no longer applies and Stealth minions will only lose Stealth when it attacks.
Stealth can be accessible in any class via Neutral cards, although it's strongly associated with the Rogue class.
Keywords that Work Great with Stealth
Stealth minions have a strength in which they can survive a turn much more likely than other minions, especially early game. Lifesteal on a Stealth minion can give you a good chance of a heal, and Taunt will give you a wall after it loses Stealth. Poisonous on a Stealth minion can also force your opponent into an awkward situation where they might not be able to deal with the Stealthed minion in question, and will later allow you to use the Poisonous ability.
Playing Against Stealth in Hearthstone
Stealth minions are fairly rare, so they usually aren't a massive problem. In the event that you are playing against a Stealth-centric deck however, have some AoE on hand so you'll be able to take out Stealthed minions even while they're in Stealth. Flare or Streetwise Investigator can also remove Stealth, although the former is generally preferred for its ability to take out Secrets.
Example Hearthstone Stealth Cards
Spymistress is a fairly simple, strong Stealth minion with aggressive stats.
Greyheart Sage is a card that specifically plays into Stealth Rogue. While it lacks Stealth itself, it will draw you 2 cards if you play it whilst you control a minion in Stealth (whether or not that Stealth is natural).
Burrowing Scorpid is a card that gains Stealth if its Battlecry kills a minion.
Valeera the Hollow is the only card in the entire game that can grant a hero Stealth. It otherwise works the same as Stealth on a minion.