There are no standard sizes or shapes for greens; they vary widely in both regards. Most common, however, is a shape that is rounded. As for green size, the greens at Pebble Beach Golf Links, one of the game's most famous courses, are considered small at around 3,500 square feet each. Greens of around 5,000 to 6,000 square feet are fairly average.
Greens have the shortest grass on a golf course, because they are designed for putting. You need short, smooth grass for putting; in fact, the official definition of "putting green" in the Rules of Golf is that area of a golf hole "that is specially prepared for putting."
Putting greens sometimes are level with the fairway, but often are raised slightly above the fairway. Their surface can include contours and undulations (which cause putts to "break," or veer off a straight line), and can pitch slightly from one side to another. Just because the green is specially prepared for putting doesn't mean you get a perfectly flat, easy putt.
You are allowed to pick up your golf ball once it is on the surface of the green, but you must place a ball marker behind the ball before lifting it. The play of a hole is over as soon as your ball drops into the cup where the flagstick is located.