Why is the selected object displaying in the 3D view, but not rendering?
In object mode, it shows up:
However, it does not show up in render mode:
Why is this so and what causes it to hide in render mode?
There are a lot of reasons why this might be happening. While these may not be the reasons for why it's happening for you, it's a good place to start.
This is one of the most common reasons for something not appearing in a render.
Just make sure that this is always checked for any objects that you want to render.
When working with Blender Internal, there are some issues with materials that could be causing your object not to show up in the render.
Having those settings in your material will cause the material to be completely transparent. Having a texture effect the alpha can also cause this.
There are a few ways an object can be accidentally hidden from a render in cycles.
Probably the most common, is to accidentally uncheck camera from the ray visibility in the objects tab of the properties panel.
Unchecking camera will basically make the object not show up in the image (although reflections, shadows, etc. may show up and the visibility of those can be controlled using the other checkboxes)
In cycles, an object could also disappear from the render, if the material was transparent (contained a transparent shader).
The last major reason why an object might disappear from a render, is layers and renderlayers. Always make sure that the object that disappeared is on one of the currently viewed layers.
For renderlayers, be sure that you have all of your scene layers selected. These are the same as your currently viewed layers, and are tied to them. If you select or deselect a layer from there, it will also be selected or deselected in the viewport.
The layer section is where you include the layers that you want used in a specific renderlayer.
The last things to check is your exclude and mask groups. The exclude group simply excludes the contents of a layer completely. No lights or anything similar from those layers will be shown.
For mask groups, just make sure that you aren't masking out any of the layers that you want visible. If you have a mask layer enabled, it will automatically cut out any pieces of it that appear in a higher Z-Depth.
Flip the direction of the normals on the torus.
You can do this by selecting the object, entering Edit Mode, selecting all vertices, and then pressing CtrlN