Here are some links and information about FDM 3D printing. The first thing you need for a 3D print is a 3D drawing. You can create your own using software like Tinkercad for beginners and children. On the professional end of the scale is Solidworks, Autocad (free version), or Sketchup. There are more just do a Google search. Autocad and Tinkercad have Chrome apps. You have to save or export your drawing to an .stl file format.
If you don't want to jump in to drawing yet you can download pre-made drawings. From websites like Thingiverse (free) or Yeggi (a search engine for 3D models), Turbo Squid. Of course the drawing programs all offer drawing in their format that can be saved or exported as an .stl file.
Next you'll need software to "slice" the .stl file. Slice is the term for generating the coordinates in X Y and Z 3D space where the printer will place the melted plastic. That information is saved in a G-Code file (.gcode). Some software for that operation is Slic3r, Simplify3D, and Cura. These can be incorporated into programs designed to give you complete control over your printer. Repetier and Matter Control are examples of printer control programs.
Of course you will need a 3D printer to bring your object into the real world. Here are some to look at. First if you are interested in the famous do it yourself Reprap Project where a 3D printer is used to print the parts for another 3D printer. After that is the Makerbot line starting with the Mini at $1375 as I write this. I decided on a Robo3D for a balance of cost $799 and features. The market place is exploding. At a recent Maker Fair in Kansas City I saw the Q3D Printer which starts at $199. I was impressed with this tiny printer.
Currently I mostly use models from Tinkercad. Slice them using Matter Control and Cura. Then load the G-Code on a SD card and print on my modified Robo 3D.
I hope this will help you get started in the fun of 3D printing.