Face

The face is a slang term for a player's hero as the target for damaging attacks and effects, as opposed to minions.

"Going face" (or go face) or "attacking the face" refer to focusing one's attacks on the enemy hero, rather than their minions. "Face damage" refers to damage to the enemy hero.

The term likely derives from the fact that heroes are represented by portraits, so to target the opponent with damage, the player must literally mouse over the enemy hero's face.

Face decks
Almost all games must eventually involve damage to the face to win, but as an overall strategy, "going face" is used especially by aggro decks. Decks that are strongly focused on dealing face damage are often referred to as "Face [class]", for example Face Hunter. Face decks are arguably the epitome of the aggro deck playstyle, with little or no cards designed to deal with the opponent's board, instead focusing purely on dealing face damage.

Strategy
Traditional decks tend to focus on things which manage minions and allow one's own minion army to thrive, as they give free damage per turn when they survive for long-term play, as opposed to 1-shot damage spells.

This is why it's often better to target minions with such spells as that deprives the enemy of ongoing free attacks per turn, while also sparing damage to your own minions in accomplishing it.