All you really need to know is the simplified answer which is in the "What do the levels mean for the words in a lesson?". So make sure you've read that before reading on here.
What follows is a very geeky answer, so you can stop now unless you're really interested to know all the geeky detail.
A card is the term given for a word or phrase in the context of moving through the system of boxes.
All cards start off as "unstarted". This is like a box of unlimited size labelled "unstarted". This starting box can even be thought of as box zero (indeed in the program it is implemented as box 0), but the website calls it "unstarted cards" which is hopefully a little less intimidating. There are 10 other boxes labelled from box 1 to box 10.
Each box has a target maximum number of cards which the box can contain. As the levels get higher, the maximum number of cards get greater. This, together with the other rules below, ensures that as cards go up the levels they are asked less and less often.
The size of the boxes is as follows:
The card to select next is chosen by the strict application of the following rules: