I'll make sure to explain this using the most technical words that I can. Enjoy :)
Step 1: Overview
Yeah, that's a boring title, but it gets to the point. 3D printing, or additive manufacturing, is a process where a material is melted in a way that it can be deposited in a controlled manner. Then, the material is quickly placed layer by layer into the shape that you want. The most common material is PLA plastic, but there is ABS plastic as well as a multitude of other materials, even a mixture of wood! 3D printing has become much more popular recently, but it has been around for a long time, often being used by companies to create prototypes without using too many resources.
Step 2: 3D Printers
3D printers come in many shapes, sizes, and costs. The cost of a printer is usually dependent on the size of the print volume (how big of a piece it can print), accuracy (how thick its layers are), and build quality. The cheapest printers right now are about $200 - $300 USD. If you are in the market for a printer, I would recommend the Makerbot Replicator or the Ultimaker 2. The Makerbot is a bit less than $3,000 and is accurate and has a great build volume. It is also user friendly. The Ultimaker 2 is about $2,800 and also has great build volume and accuracy (it can print a layer 1/5 the width of paper). If you have done your homework and know a little something about 3D printing, then buy the Ultimaker. Of course, this is pure research because I do not even have a 3D printer...
Step 3: How Do I Print Something?
Well, first you have to know what you want to print. To tell the printer what to print, you must create or download something in a CAD (computer aided design) program. For kids I would recommend Tinkercad. Then, once you have your file (it should be .stl), you put it on a flash drive or SD card and put that into the printer. From then, it depends on the printer.
Step 4: Creating Something With CAD
CAD, or computer aided design is mandatory for 3D printing. There is an X, Y, and Z axis to give it the 3 dimensions that you need. Again, CAD programs can differ greatly, so I can't really help you here, but the CAD program should have a tutorial. If not, go watch a tutorial online. Or, you can just download something from sites like Thingiverse, which you can download files to your CAD program. Once you have your model, save it onto a flash drive or SD card as an .stl file. Try to make sure that the model does not have many overhangs that the printer cannot perform.
Step 5: The Print
Now that you have put your flash drive in your printer, start the print (this differs for different printers). For a 3D printer to work correctly, it must have filament. Filament is usually on a spool and usually made of plastic. The filament travels through the printer into the extruder, which moves around the build platform. The extruder consists of a fan, which blows on the filament that has just been laid down, a motor, which pulls the filament from the spool, and a hot end, which heats up the filament to its melting point so that it can be placed down.
Step 6: Completed Print
Once you have completed your print, take it off of the build platform (may take a little force) and you now have your item that you designed in your CAD. It seems like science fiction, that you can take something from your computer and have it in your hand in a few hours. Congrats, you have completed your first 3D print. Tinkering with settings and other things will expand your knowledge of 3D printing.