Airplanes built by their owners are called by a variety of names: amateur built, home built, experimental, and others. I and many other people prefer "custom-built" because you're building the plane yourself, for yourself. Within certain limits you can modify the plane to fit your own needs and tastes. It's your personal aircraft!
Why would anyone want to build a plane? Many reasons, actually. By doing the work yourself you will save lots of money developing what's often called "sweat equity." You'll also know exactly what you're getting as you put each piece together; you'll probably be even more careful than the aircraft worker assembling a plane for an unseen customer. One more big reason: You'll be able to legally do much of the maintenance and repair on your aircraft, saving you big bucks.