I am by no means an expert on this but seeing as no one has answered as of yet, I'll give it a try.
The chance of receiving a card is #/20, where # stands for the number of the waves that you have currently completed. So if you have done 10 waves, the chance is 10/20 = 1/2.
Only one of each card can be owned at a time, so my guess is that if you own 3/4 cards, you will be guaranteed the 4th card IF you are awarded a card at all. In other words, if you have 3 cards and complete 10 waves, you will have a 1/2 chance of receiving the fourth card. If you complete 20 waves, then you should be guaranteed to get the fourth card.
Each card increases the chances of receiving specific drop from that monster. I'm not sure about what the frequency, but the drops that have an increased chance are the following (with links):
Abyssal demon - Abyssal whip/abyssal wand
Glacors - Glaven boots, Ragefire boots, or Steadfast boots.
Tormented Demons - Dragon claw, Off-hand dragon claw, or dragon limbs.
Automatons - Pneumatic gloves, Static gloves, or Tracking gloves.
Another thing to note is that the card turns to dust after receiving a unique drop (i.e., one of the items that I've just listed).