You can see what special skill a card has by touching スキルせつめい (explain skill) for Rider card and カードこうか (card ability) for special card.

For Rider cards, generally (for the first 2 packs i've unlocked, anyway) the ability is formatted like:


which means:

'When this card is in (position), if (requirement) is fulfilled, (stat up/down)’

The position is either
ゼンエイ (front guard, scanned first) or
コウエイ (rear guard, scanned last)

Sometimes, the first requirement is not dependant on the card's position, but the card's partner and position.
For example, 2-004 Decade's requirement is コウエイがキバのとき (which is "When Kiva is in rear position...")

The common requirement is:
なかまとゾクセイがおなじだと - if this card have the same attribute as its partner
あいてとゾクセイがおなじだと - if this card have the same attribute as the enemy
or sometimes there's no requirement at all.

The status capable of getting bonus is:
タイリョク - Life/Stamina
コウゲキ - Attack
ボウギョ - Defense
ヒッサツ - Special Attack
アタックポイント - Attack Point (I think this refers to the numbers on the slot)
Sometimes, for cards with skill that activate during rear position, they add "なかまの” which means "Partner's...", since the rear card's status is not taken into account.

So, for example, 2-007 Kuuga's skill description is
コウエイのとき (When this card is in rear guard position)
あいてとゾクセイがおなじだと (If it has the same attribute as the enemy)
なかまのボウギョ+200 (Partner's DEF + 200)

Special cards generally have generic stat up/down effects, but some of them are unique.