A doughnut is, in fact, pretty much a torus. At least, it would be if its cross section were a perfect circle.
Wikipedia defines a torus as "a surface of revolution generated by revolving a circle in three–dimensional space about an axis coplanar with the circle." This is easier to visualise (for me, at least) the other way round: imagine a vertical cross–section through a ring doughnut, and assume that it's a perfect circle. Now imagine that circle revolving around the centre of the doughnut; it would trace a doughnut–shape in space. This is the "surface of revolution".
A toroid is the surface of revolution formed by revolving any closed curve about a coplanar axis. For example, if instead of a circle you had a non–circular ellipse, the surface of revolution would be a toroid but not a torus.
Also according to Wikipedia, a doughnut is actually what's known as a solid torus. To make a solid torus, you rotate a disc (i.e. a solid two–dimensional figure, not a curve) around the central axis.
An inner tube is a torus – but obviously not a solid one.
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