Once you have your STL or mesh, you need to run it through a slicer. A slicer takes your print settings and like a deli counter slicer, it slices it into individual 2D layers that your printer will receive as commands. The slices can be thick or thin, depending on your settings, and also controls how fast your printer will print each layer.
123D Make (this app lets you take a 3D file and re-create it using cut cardboard or paper) does a great job of illustrating what a slicer does. Below is a 3D scan of my assistant Adam as an STL mesh.
Now I can slice Adam in 123D Make and get a glimpse of each layer:
If I were going to recreate Adam in cardboard, I can see what each layer of that creation would look like, this is similar to what slicing software produces for a 3D printer:
Common Slicing Software
The output of the slicing process should be a file that contains all of the commands that will be sent to your printer in order for it to create the object you have designed. This is usually a GCODE file. You can then load this GCODE file into your Printer Control Software and start your print. If the slicing process fails though, you may need to repair your mesh with a mesh repair tool before it will slice correctly. You can also try a different slicing tool as sometimes other slicing tools will handle issues with a mesh differently.