I've heard the term "press high" repeatedly while watching the World Cup. What does it mean? Do teams that press high generally enjoy more possession or less?
It means the team without the ball, moves higher up the pitch to put pressure on the team with the ball, with the idea of getting the ball back or forcing the other team to make a mistake.
Say, teamA has the ball near their own goal. Then teamB will start putting pressure on teamA, by moving higher in the direction of teamA's goal. Players of teamB will start attacking teamA players to get the ball back. But the key point here is, it will be done nearer to teamA's goal. That's why it's called pressing high.
As shown in the image below, blue team tries to pressure the red team, near the red team's goal.
Few benefits of pressing high are:
A disadvantage would be, you are more susceptible to quick counter attacks, and over the top balls.