I'm glad you asked. I'm going to describe my current method for shaping a seat, though it is important to note that furniture makers have been shaping seats for centuries using traditional chair making tools. One of them, an adze, is sort of like an ax except the blade is parallel to the surface you are cutting instead of perpendicular. Also, the blade is curved. You can see other tools such as scorps and travishers (I love those words!) at the bottom of this page.
The current method I use to make seats involves shaping individual pieces of wood and gluing them together. I usually use three pieces. Here's how I do it: