Card backs are a purely cosmetic feature in Hearthstone that allows you to customize the underside of your cards with varying designs. These are displayed at all times on the right side of the board where both your deck and your opponent's deck are shown with the preselected option. Your opponent also sees your card back at the top of their screen representing the cards in your hand. They also sometimes appear outside of matches such as when you're opening packs. 

All of the card backs can be found in your Collection by clicking on the card back icon to the right of the class tabs. By right clicking on a specific one, you can denote it as "My Favorite" which will apply it globally to both decks you have created and all you create in the future. This can be changed at any time. There is also the ability to assign them by deck. Doing so requires you to open a deck and go to the same card back tab. Reassigning is as simple as clicking and dragging your new selection over your old one.

There are a variety of ways to collect new card backs but the most common are by reaching Rank 20 in Ranked Play during the course of a season. Each month Blizzard designs a new one meaning there are 12 per year with a total of 19 thus far. These are currently not obtainable after the conclusion of a season, but Ben Brode has commented that this will likely change at some point.

Currently Obtainable

Here's every back that you can currently unlock in Hearthstone.

Free

Name Card Back How to Earn It
Classic Granted to all Hearthstone accounts.
Legend Achieve Legend rank during any single season.

June 2016 Ranked Reward

Awarded for reaching Rank 20 in Hearthstone this month.
Name Card Back How to Earn It
Zul'Drak Achieve Rank 20 in June Season 2016 (Ranked Season 27)

Requires Purchase

To unlock these you either need to spend gold or actual money to acquire them (varies by card back).
Name Card Back How to Earn It
Heroic Naxxramas Purchase Naxxramus Adventure and complete it on Heroic.
Nefarian Purchase Blackrock Mountain Adventure and complete it on Heroic.
Magni Purchase Magni Bronzebeard.
Medivh Purchase Medivh.
Warlords Purchase World of Warcraft: Warlord of Draenor Collector's or Digital Deluxe Editions.
Legacy of the Void Purchase Starcraft: Legeacy of the Void Collector's or Digital Deluxe Editions.
Overwatch Purchase Overwatch Collector's or Origins Edition.

Promotional

Most of these require you physically being somewhere with other Hearthstone players.
Name Card Back How to Earn It
Fireside Play three games against players on the same subnet
with at least three players total on it at the time.
Dalaran Flame Be an active member of a TeSPA (The eSports Association) local
chapter with at least 25 members.
Was also granted to Heroes of the Dorm attendees.
Heroes of the Storm Reach level 12 in Heroes of the Storm.
Samsung Galaxy Play Hearthstone on a Samsung Galaxy S6 series phone.
Power Core Play in or attend select Hearthstone tournaments around the world.
First rewarded at the EU Road to Blizzcon finals in Prague.

Retired

Ranked Rewards Card Backs

Head over to our Seasons database for information on the rewards for each respective season.

Promotional

Name Card Back How to Earn It
Blizzard 2014 Buying a physical or virtual ticket to Blizzcon 2014.
Blizzard 2015 Buying a physical or virtual ticket to Blizzcon 2015.
Molten Core Preordering the Blackrock Mountain adventure.
The Grand Tournament Preordering the 50 Grand Tournament packs bundle.