From a construction standpoint, both hand woven and machine woven rugs, with good fibers (wool!) and good dyes, are going to be good rugs.
What are plush rugs made of?
Shag rugs come in a variety of materials, including wool, faux and real fur, leather, and acrylic. They also come in more popular rug materials like polyester and polypropylene. A higher pile looks warmer and cozier, but also requires a little more maintenance.