Greens can vary widely in shape and size, but are most commonly oval or oblong in shape. They can sit level with the fairway or be elevated above the fairway. They can be flat, sloped from one side to the other or contoured all around their surface. In other words, there are no hard-and-fast "rules" about what size or shape or other design elements a putting green must have. What a green looks like, and how it plays, are up to the course designer.
The rules of golf define putting green as any ground on a hole "that is specially prepared for putting." "Specially prepared for putting" means that the putting green has the shortest turfgrass found on a golf course, and that grass should also offer the smoothest surface found on the course.
In the Rules of Golf, Rule 16 is dedicated to the putting green.