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Generally speaking all cards activate once and once only when you play them. You resolve all of a card's effects (including gaining recruit and attack points) then move on.
Hopefully this post I wrote a while ago may help : https://boardgamegeek.com/article/19596654#19596654
To save you a click though, the most important part is this:
Three types of "effect" exist on the card descriptions - note that some cards carry more than one of these :
<Effect> - standard effects
These things you'll get whenever you play the card, no conditions, just like the base attack and recruit points and you get the effect once*.
<Symbol> <Effect> - super power effects
You activate this if you have previously played a hero card with the associated symbol (whether that be faction affiliation or power-type) but regardless of the number that have been played, you get the effect once*.
<Symbol> <Symbol> <Effect> - critical strikes
Added with Dark City, you activate this if you have previously played the requisite hero cards to trigger the power, usually two of the same type, but occasionally two different power types but regardless of the number that have been played, you get the effect once*.
Note that there are four key exceptions to the "you get the effect once*", hence the asterisk:
"Whenever" : These abilities will trigger each and every time the trigger conditions are met after you have played the card. Exceptional exception: some of these cards trigger in other ways, it'll be pretty clear on the card, however. But remember that you can trigger them as many times as necessary.
"May" : These abilities allow you to choose not to carry out the effect. This can be very helpful (for example, choosing not to use the Shadowed Thoughts effect is often preferred in tough schemes).
"For each XYZ hero you have played this turn." : These abilities will carry out their effect once for each card of a given type that you have already played this turn. They tend to be some of the most powerful cards in the game, assuming you can trigger them properly.
"For each XYZ hero you have." : These abilities will carry out their effect once for each card of a given type you have at the time you play the card whether they're played in front of you or in your hand. This is an important distinction and could be very powerful, except that the triggering conditions tend to be rather limited (and the benefits rather minor).
N.B. That top bit is no longer completely correct, there are now a few other weird things that trigger effects based on special keywords such as Savior and Patrol the _____ but once you've got the basics down you'll find it easy to work those out.