Not to be confused with support characters (fighters that mainly aid teammates rather than doing damage themselves).
Support cards are cards that can be bought with Power Credits or Alliance Credits to increase the strength of characters. Support cards bought with Power Credits (also known as Support enhancements) give their bonuses (+10% health, damage, or energy generation, which is not to be confused with power generation) to all versions of the same character, while Alliance support cards gives their bonuses (+3%~7% health or damage) to all characters. Some support cards increase experience gain or energy generation for all characters.
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The Support Cards icon in the Store.

Support Cards can be bought with Power Credits and can also be randomly received through buying Bronze Booster Pack, Silver Booster Pack, Gold Booster Packs, and Challenge Booster Pack. In this case, the term "support cards" also includes Special Attack upgrade cards.

Support Cards can also be recieved by buying early access packs, per the 2.8 update. 

Presentation 2

An example of the received support cards in an early access pack.

Improving characters
 Level Promotion Support cards Augment Gear