Evolving is the process of combining two cards to create an upgraded or new card. Candidates that can be combined with a card can be seen under Cards > My Cards > Select Card and scrolling to the bottom where the details are located.
Evolving Normal (N) CardsEdit
Combining two identical Normal cards that are not fully evolved (the circles in the bottom-left corner of the card are not all filled up) will result in a directly stronger card, adding another "+" in its name. For example, combining Archer+ and Archer+ will result in Archer++.
Evolving H-Normal (HN - High Normal) CardsEdit
If anyone has any information, please feel free to help to contribute. It does say that you can evolve HN cards when you select one of them to enchant. A message pops up saying "HN cards must be maxed before evolving."
Evolving Rare (R) and Super Rare (SR) CardsEdit
Not all Rare and Super Rare cards can be evolved; however, the ones that are able to work the same way as normal cards. Two identical Rare cards are needed to be combined and evolved (upgraded) into a stronger card, ending with a "+" in its name. For example, Conjurer and Conjurer can be combined to evolve into Conjurer+.
Stat Gain From EvolvingEdit
Evolving Super-Rare cards seems to grant +40 Atk, +580 HP. (Based on Ocamel; haven't examined all of them)
Evolving Ultra-Rare cards seems to grant +20 Atk, +290 HP. The bonus is less, but you get better carryover stats to partially make up for it.
If you use leveled cards as the input to your evolution, you get "carryover stats". For example, a level-5 Gunner has 78 Atk and 856 HP. If you evolve two level-5 Gunners together, you get carryover stats equal to 7 ATK and 85 HP, which are added to the base stats of the resulting level-1 Gunner+, for a total of 85 ATK and 1033 HP. In general, when evolving max-level cards, the most common result seems to be a carryover bonus of equal to 10% of the parent cards' stats. If either of the input cards is not max-level, you get a worse result.
This forum post: http://m.gamevilusa.com/forums/showthread.php?93250-Guide-Carryover-Stats-Guide has a message from the developers on the topic. The main unexplained comment is that the carryover stats are 10% for some cards but up to 20% for other cards (maybe super-rares carry over more?).
Bonus stats DO carry over from previous evolutions, but the effect is very small. For example, if we took two of our special Gunner+ cards from the example above (which already had carryover stats), and we leveled them to 5, they would have 93 Atk and 1057 HP. If we then evolved them together, the resulting Gunner++ would get carryover stats equal to 10% of the parents' stats, or +9 Atk, +105 HP. For comparison, if we'd used two non-special Gunner+ cards and leveled those to 5, they would have only 84 Atk and 972 HP. If we evolved those Gunner+ cards together, the resulting Gunner++ would get carryover stats of +8 Atk, +97 HP. The difference is only +1 Atk and +8 HP, which is pretty lousy considering all the work we put in to level four Gunner cards to 5. But it does exist.
Adding equipment does not increase carryover stats.