The simplest way to explain these are the fabrics you find in those cozy winter tracksuits, hoodies, jackets.
These may also be called jogging fleece, active fleece, traksuiting material, and french terry fleece. Especially on the right side, it is the same as your usual french terry fabric, smooth and either printed or in a solid colour. On the back, the material is lined with a layer of fleece, which can range from thin (a brushed surface) to super thick (like a separate layer of sherpa fleece).
Composition wise, sweatshirt fleece is similar to french terry: mostly cotton, 3-5% of spandex/elastane, and 2-5% polyester for the fleecy part.
Project ideas: hoodies, pullovers, jumpers, jackets, loungewear and of course a good old pair of trackies (i.e. sweatpants).