Fairways are the connections between teeing grounds and putting greens. The grass in the fairway is mowed very short (but not as short as on the putting green), and fairways are often set off and easy to see because of the contrast between the height of grass in the fairway and the taller grass - called the rough - on either side of the fairway.
The fairway doesn't promise a perfect situation for your golf ball, but keeping your ball in the fairway as you play toward the green does vastly improve your odds of finding the best playing conditions.
Fairways are typically maintained by groundskeepers, mowed, manicured, in many (but not all) cases watered; as opposed to those areas of the course on either side of the fairway, the rough, which might be unmaintained or minimally maintained.
As you stand on the teeing ground of a par-4 or par-5, your goal is to hit your ball onto the fairway, advancing the ball toward the green, avoiding the danger of the rough, and giving yourself the best chance of success on your next stroke. (Note that some par-3 holes do have maintained fairways, but many do not because, as previously noted, the goal on a par-3 hole is hit the green with your first stroke.)