A divination card is a type of item that can be collected to exchange for a reward. There are many different cards, each one dropping in a different part of the game and granting a different reward.
This game mechanic was introduced to give more deterministic approach when farming for specific poe items. Before divination cards, every item drop was completely random and some items were extremely hard to get as a drop yourself. Now you can farm for specific divination card set and redeem it for specific items.
Character level is never considered to be higher than 68 in regards to drop rate penalty, meaning that high level characters (68+) can farm world areas in later acts without severe drop rate penalties. For example if the monster level is 66 there is a 0% drop rate penalty, and in zones with level 56 monsters there is a 25% drop rate penalty.
There are almost 200 different divination cards in the game, some extremely rare and some extremely common. One key aspect of Divination cards is that they drop in specific locations – for example, if I wanted to farm divination cards for Tabula Rasa (Humility cards), I’d have to continuously kill monsters in Blood Aqueduct or Channel Map.
One more thing about divination cards that different cards have different set size – that means that some sets need more cards to be completed than others. Only a full set can be traded for the item referenced in a divination card (some sets are full at 1 card, some sets are full at 16 cards).
PoE Wiki is a great resource for Divination cards – you can see where to farm them and how many cards are in a complete set.