Starting with the fundamentals, the art and science of 3D printing is simply about printing things...in 3 dimensions. No shocker there. Basically, you start with a model, which is the word we use for a 3-dimensional shape saved in a digital file. That model gets processed and turned into instructions for the printer. When the 3D printer gets those instructions, it can use a material like plastic to actually build out the shape of the model, turning it into a solid 3D object.
So where do the models come from? Most people find them online. There are websites like thingiverse.com and thangs.com which are like search engines for 3D models. Designers share their models there, and people can search for free designs to print. Even if you don't have access to a 3D printer, you can hop on to a site like this and browse the millions of models people have designed and shared. Some professional artists sell their 3D model files directly, which lets people buy licenses for models and print them from their own printers. And of course, if someone has the tools, skills, and inspiration, they can create their own models to print. (Which is a super cool power trip for creative people.) Most designers use 3D design software to create their models. While they are still kind of rare, 3D scanners are another way for people to make their own models by using a special "camera" to capture the shape of a real world object and turn it into a model. 3D scanners are still a little pricey for a lot of people, and kinks are still being worked out of the technology (But if anyone wants to sponsor a review... : D ).
The next step is taking that model and translating the shape into a set of commands that the printer can understand. This is almost always done by taking the model and “slicing” it into horizontal layers, all the way from bottom to top. The slicer basically turns every layer into a 2D drawing or image, puts the layers in order, and tells the printer how to “draw” each layer on top of the one before it. So, another way of looking at a 3D print, is to see it as a stack of 2D layers that were drawn out in plastic(or resin or whatever…) one at a time.
Once you have the instructions for the printer, you send them/copy them to the printer and then patiently let the printer do all the work.