Until recently, printing -- on clay, papyrus, cloth or paper -- was always a two-dimensional process. Then, starting in the 1980s, various technologies evolved to add the 'z' axis, which allowed machines to build three-dimensional objects from a CAD model or a 3D scan. The original general term for this process was 'additive manufacturing', and the canonical use case was 'rapid prototyping' -- the production of a plastic, wax or even metal object as an integral part of the product design workflow. Nowadays, '3D printing' is the umbrella term for these and other similar activities.
Consumer 3D printing has received a lot of breathless coverage in recent years, which is why it now sits firmly atop the Peak of Inflated Expectations in Gartner's 2013 Hype Cycle for Emerging Technologies. However, the much longer heritage of enterprise 3D printing (or additive manufacturing) sees it climbing the Slope of Enlightenment towards the Plateau of Productivity: in this market, the bulky and (very) expensive machines of the 1990s have got smaller, cheaper and faster, and evolved to handle a wide range of materials, making them a key component of many industrial design departments.
Also notable in the same diagram is the rise of 3D bio-printing, which has already seen living tissues combined with a 3D-printed 'scaffold' to create working replacement organs. Expect to see a great deal of development, and media coverage, in this area in coming years (see for, example, TechRepublic's story on the project aiming to bio-print a human heart).
Our Dipity timeline hopefully gives a flavour of the developments in enterprise and consumer 3D printing, and 3D bio-printing, over the last 30 years. As ever, do let us know if we've missed any key technologies, companies, products, services or applications.