Cards are used in Expedition mode
First and foremost; cards do not do anything, throughout the single player game. They are used to add conditions to the expedition game mode, which in modifies the multiplier applied to your end score.
To explore this feature, select "Expeditions" from the main menu. After selecting the mode and challenges to apply to the expedition, you are given the opportunity to add cards, before starting the game.
Most of it is self explanatory; however, I will still go through an example, just to cover all the bases.
An example from the guys at eurogamer
For example, the above screenshot shows a hand of 5 cards. Each card has a multiplier of +100%, and if the user completes the expedition, they will receive another1 5x multiplier to their score. Here are the five cards, in more detail:
One life to live - Dying ends the replay immediately.
Grip - Hold X to hang from ledges.
First Aid - Health regeneration is disabled. Use bandages to heal.
Armor-Plated - Enemies have double their base armor strength.
Prize Fighter - Deal more melee damage, but bleed when damaged by melee attacks
Understanding the Values
The cards each have three separate values, detailed at the top of the card. This helps ascertain the net value and benefits of using the card.
This tells you how much the card will sell for, if you decide to cash it back in for more credits. Useful if you do not intend to use a card, as you can effectively recycle unwanted cards towards potentially better cards.
This is what you will be paying attention to, a majority of the time. This value will tell you how much of a points bonus the card will give. Assuming you are trying for a high score, your best strategy is to raise this number up as high as possible, given your available cards. This is often2 also an indication of the difficulty the card may add to the game, in offset of the extra points.
1While unrelated to the context of the question, there are other modifiers that can be applied before the card selection screen, that will also add to your multiplier. For example, adding challenges, and selecting harder levels of difficulty.
2Not always the case; The grip card is of much higher value, as it gives the same bonus, without the rise in difficulty. Other modifiers to difficulty may be 'non events', due to personal strategy; for example, only having one life may not be an issue if you can smash through the expedition on your chosen level of difficulty, and any disadvantage in melee fighting may be circumnavigated by taking the stealth approach.