Modeling Software for 3D printing

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3D modeling software seems to take one of two perspectives: Artistic or Engineering.

Artistically oriented

Blender

https://www.blender.org/

The open source gem of the 3D modeling world.  Primarily aimed at 3D visual modeling and animation, it  can also be used to prepare 3D printable models.  Making 3D-Print-ready STL files requires some care.  Enormously rich and complex user interface, but much can be ignored (shading, animation, etc), if you're only interested in creating printable models.  Tons of info online.

MoI

Moment of Inspiration is a relatively inexpensive 3d modeler that runs on PC, Mac, and Linux. Less full featured than Blender or Inventor, this makes it a lot easier to learn. It produces great STL files as long as you are careful that your exported objects are solids (see the FAQ about manifold).

Engineering oriented

OpenSCAD

http://www.openscad.org/

OpenSCAD implements a programming language for 3D modeling.  There's little it can't do, but you really have to be a computer proramming nerd to use it.  If that's you, have a look; you might like it.

SketchUp

http://www.sketchup.com/

Formerly 'Google SketchUp', this 3D modeling tool has been around a few years and has developed a strong following.  You must install a plugin to export STL files.  Be careful: the exported STL files can have flaws that must be cleaned up prior to slicing.

FreeCAD

http://www.freecadweb.org/

FreeCad is an solid modeling CAD program with a modeling paradigm much like Autodesk Inventor or Solidworks.   It's pre-beta and has bugs and incomplete documentation, though a quick web search usually finds anything).  Nonetheless, it's quite capable.  Your author is a big fan and has never seen it produce a flawed STL file.

Solidworks

http://www.solidworks.com/

A very popular, professional 3D modeling tool.  Very expensive.

Autodesk Inventor

http://www.autodesk.com/products/inventor/overview

Autodesk's answer to Solidworks.  Also very expensive.

Autodesk Fusion 360

http://www.autodesk.com/products/fusion-360/overview

Autodesk's cloud-based modeling tool.  Very capable and free for hobby use.  We frequently host classes on Fusion 360, taught by Autodesk's own instructors.