The typical, consumer-grade 3D printer creates objects made of thin layers of stiff, brittle plastic fused together. Of course, hard plastic isn't ideal for all projects, so that's why researchers from Disney, Cornell University, and Carnegie Mellon Univeristy have developed a new 3D-printing technique that creates objects out of layers of felt.
Disney's fabric 3D printing method starts by taking a 3D model of an object, and "slicing" into printable layers--a typical part of the 3D printing process. Next, the printer laser-cuts shapes out of adhesive fabric that correspond to the sliced layers, then transfers that layer onto the printer's build platform. It then applies heat to each layer to "activate" the fabric's adhesive.
The printer repeats this process to cut, stack, and adhere each layer until it completes the model. TechCrunch characterizes the printer as being "as much a laser cutter as it is a 3D printer." The output is a bit rough, but it's an impressive process nonetheless.
Seeing as this is a research project, you shouldn't expect to be able to purchase a fabric-based 3D printer any time soon. If you would like to 3D-print a soft, squishy object, though, you can submit your 3D model to Shapeways and have it printed using that company's Elasto Plastic material.