Different software packages/services that can be used to build your own custom 3D models for printing.
TinkerCad - An online, browser-based, modeling environment. Tinkercad is a product offered by the heavyweight of the industry: Autodesk. While free, this tool is also a less capable/less powerful tool than paid products.
Fusion 360 - A desktop modeling environment. A much more powerful option than TinkerCAD, its also got a much steeper learning curve. Like Tinkercad, it is published by Autodesk. While offered as paid product for commercial applications, there is a free version for hobbyists and (very) small businesses.
AstroPrint - Astroprint's purpose is to provide services to allow you to cloud print to 3D printers. Part of their software package allows for slicing STL files in their cloud. Once sliced, you can then download the completed G-Code file for transfer to a 3D printer. If using TinkerCad on a Chromebook, you will need to use this or similar service to slice your model to prep it for 3D printing. *One caveat, there is a reccuring issue with the site where options aren't there to start with, then later the user comes back, and the option is there. So, if you can't find the slice option, or the profile for your printer, go do something else for a couple hours and come back to the site later and try again.
Ultimaker Cura - Ultimaker Cura is software for producing the CNC gcode for the printer from the 3D model. This process is called 'slicing' and produces the tool paths for the extrusion head of your printer. Requires installation on either Windows, MacOS, or Linux -based computer.
While today many businesses provide 3D printing services, the idea at the onset was that people would be able to design up models and share them freely, "give something, get something" mentality. This started with the RipRap printer, which could be used to print its own upgrades, as well as many of the parts (non electrical components) to build a second or even third printer.
Several repositories do exist that allow people to share 3D models with the world. Here are a few we've found.