The word panic button refers to the phenomenon that every class in the game has some sort of a button they press in order to save their hides from a certain death. Some classes are more 'blessed' with these buttons than others but it is usually evened out by their surviving/damaging skills.
Panic buttons are nearly always instant spells with a certain cooldown, to make them available for use at all times, but not limitlessly. These abilities are usually neither Crowd Control methods nor Snares, though the Druid's 
[Nature's Grasp] is useful as a Crowd Control method and some of the other abilities have other offensive uses.
"Save yourself" buttons[]
- Death Knight: Raise Dead followed by
[Death Pact], 
[Icebound Fortitude] - Druid:
[Travel Form]/ [Aquatic Form], [Nature's Swiftness], [Nature's Grasp] - Hunter: [Disengage], [Feign Death], [Scatter Shot], [Aspect of the Cheetah].
- Mage: [Blink], [Ice Block], [Frost Nova], [Blazing Speed], [Invisibility]*.
- Paladin: Bubbles, [Lay on Hands].
- Priest: [Desperate Prayer], [Pain Suppression], [Psychic Scream].
- Rogue: [Sprint], [Evasion], [Vanish], [Cloak of Shadows].
- Shaman: [Ghost Wolf], [Ancestral Swiftness], [Earth Elemental Totem], ( [Reincarnation]).
- Warlock: [Soulshatter], [Howl of Terror]*, Sacrifice, [Death Coil].
- Warrior: [Shield Wall], [Recklessness], [Retaliation], [Last Stand], [Intimidating Shout].
(*Howl of Terror can become instant for Affliction warlocks, and Invisibility can become instant for Arcane mages.)
"Save the day" Buttons[]
Usually used to prevent the death of every person in your party/raid and having to restart all over again.
- Druid: [Rebirth], the "Battle-res"; [Tranquility]
- Shaman: [Reincarnation]
- Paladin: [Blessing of Protection] (only blocks physical damage)
- Warrior: [Intervene]
- Hunter/Rogue/Mage: Use of [Feign Death]/ [Vanish]/ [Invisibility] respectively in combination with resurrecting engineer items such as [Goblin Jumper Cables] or [Goblin Jumper Cables XL] on a party member capable of ressing the rest of the group can save the day.
For more information, see Wipe.
When and how[]
Most buttons have a pretty short cooldown, enabling it for use every one or two battles if/when needed. However, some have the potential to be used to save a lot more than you'd think so think twice before using these for PVP situations.