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
- 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], ( [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: [Divine Intervention], [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.