A chain is a row of boxes that each have two sides drawn, joined end to end. Open one and your opponent can claim every box in it.
A chain is a run of connected boxes that each already have two sides drawn. When someone draws a line into one end, the first box has three sides, the next box falls to its fourth side, and so on down the row. The player who did not open the chain can take every box in it.
Loops
A closed ring of such boxes is a loop. It works the same way but costs the player who opens it a little less, because the last four boxes cannot all be taken in a straight run.
Why chains decide games
Because you move again after each box, a long chain is worth many turns at once. Players avoid opening one for as long as they can, which is why short chains go first. The strategy guide covers counting safe moves and giving up the last two boxes of a chain on purpose.