A free card is, as its name suggests, one that does not cost a player (or players) anything to see. Typically, everyone remaining in a hand might check on seeing the Flop, in which case the Turn will be a free card.
Not surprisingly, getting to see a free card can be more important an aspect of poker than many might think. Whether or not it is possible to see a free card can make the difference between continuing in the hunt for the pot or, in the case of the opposition making it too expensive, having to fold.
For example, holding 7h8h with a Flop of 9d Kc 3h a player wouldn’t be justified in making any further investment to the pot. However, if nobody bets and the free card that the Turn brings is the 6h, then the hand is suddenly transformed to one that now has a flush draw and an open-ended straight draw.