3D Printer Ecosystem

Software Ecosystem

Solidworks is a very powerful commercial 3D modeling and design package.
Autodesk Inventor is a powerful commercial 3D modeling and design package.
FreeCAD is open source and crash-prone, but has a feature tree like most grown-up CAD, and has enough features to get work done. It can import 2D drawings from STL files, which Dr. Lawlor uses with Inkscape.
Any decent CAD package can import or export IGES or STEP format, which directly represents all the curved 3D surfaces of your design.

OpenSCAD is a programmer's modeling package, where you write code to make shapes, and immediately see the resulting shape. Unlike classic parametric CAD, it's possible to structure the code nicely, with named parameters, loops, libraries, and modules (like functions). However, OpenSCAD can run into numerical issues with adjacent (not quite overlapping) shapes, and does not directly support extruding or cutting relative to a surface, or filleting or chamfering edges.
Tinkercad is a very simple to use drag-and-drop 3D modeling program that runs in a web browser. It supports CSG union and difference operations (via grouping and "holes"), but doesn't support intersections, fillets, or chamfers, and reworking an old design requires a lot of ungrouping and regrouping.
Google SketchUp is a draw-and-extrude 3D modeling program. The free version makes it fairly difficult to convert to STL file (I've done it by exporting KML, manually unzipping, importing .dae/Collada into Blender, and saving STL)
Blender is an open-source 3D modeling program, excellent for organic shapes, although not designed for CAD or 3D printing, so it is easy to make unprintable objects (not closed, intersecting, etc), and it is difficult to use exact measurements.

STL files define a model, as a list of triangles around the outside of a closed surface. They're the de-facto standard for 3D printable models, although they're not very friendly for shape modification.

NetFabb Cloud can repair an "almost closed" STL file.
Shapeways, Sculpteo, or The UPS Store can print your STL file and mail you the resulting object.

slic3r is my favorite, since it has a nice GUI, but is still fast and highly customizable, and allows you to rapidly switch between different groups of settings.
Skeinforge is the old standard, a powerful open-source slicer with a bewildering interface.
KISSlicer is a closed-source but powerful slicer. It has excellent support material generation.
NetFabb is a commerical integrated software package that does modeling, slicing, and printer control.
Cura is the Ultimaker slicer.

Printer control software is the front end for operating the machine. On many commercial machines, the slicer and printer control software are merged, with varying degrees of success.

Printer firmware reads gcode and controls the machine. Often you can improve the print quality and mechanical performance by tuning the velocity, acceleration, and jerk parameters.

Marlin seems to be a popular open source firmware, although there are hundreds of customized flavors of it. It is uploaded to the printer via the Arduino IDE.

Marlin will automatically load and print "auto0.g" from an SD card at startup. This is my favorite way to run a printer without an attached computer.

Design styles:

Cartesian printer has three orthogonal XYZ axes. The bridge style (e.g., Mendel, Printrbot) has the head moving in X and up in Z, and the bed moving in Y. The box style (e.g., Ultimaker, Makerbot Replicator) has the head moving in X and Y, and the bed moving down in Z.
Delta printer has three vertical axes with linear slides.

Parts of a typical 3D printer:

Hotend extrudes the melted plastic. Hotends come in a variety of styles, but they all basically shove plastic filament into the hotend using some variety of toothed drive bolt.

Direct-drive hotends have the drive bolt directly on the stepper motor. Makerbot is famous for these, although it is difficult to reliably get enough torque at the stepper head.
Geared hotends, like Wade's Accessible Extruder, use a gear to increase torque at the drive bolt.
Bowden tube hotends separate the drive stepper from the hotend using a long thin flexible tube, which the filament is pushed through. This makes the moving hotend much lighter weight, allowing faster moves, but pushing filament can be error prone.

Bed or build platform is where the part sticks. Often heated, to soften the plastic so the part sticks better, and there's less thermal differential to cause parts to curl.

Bare glass is cheap and smooth, but needs to be heated or coated to get decent adhesion.
Kapton tape provides a compatible plastic surface to improve adhesion to a heated bed.
Glue stick or hairspray can be used to help parts stick to the bed.
"ABS Juice" (ABS + acetone slurry) can also help glue down parts.

Stepper motors move each axis: most axes are connected to the steppers with belts, but the heavier Z axis is usually moved using a threaded rod, called a leadscrew in this context. In CNC cutters, all axes are typically on leadscrews (often more accurate ballscrews), which makes them slower but more powerful and accurate.
Control board, like the Melzi all-in-one, has motor drivers for the steppers, FETs to switch the hotend, bed, and cooling fan, and inputs for the thermistors on the hotend and heated bed, and end switches.
Power supply supplies clean DC power to the entire printer. It may need several hundred watts for a heated bed. Typically this is either a 12V PC power supply, or a 19.2V or 24V laptop power supply.