Most importantly: It's not just the disks rendered from a point source, even though that's the simplest way to describe it and see it. The disk is just a shorthand; the lens characteristics that produce these disks are always present; they're what determines the look of the out-of-focus areas in every photo you take! A particular example of this is that lenses with bright-edged bokeh will also tend to show straight edges as parallel lines.
Shape: The shape of the disk is determined by the shape of the aperture, and is well-described in other answers (though see also the note on 'distortion' below).
Brightness: The brightness across the disk is influenced by other aspects of the lens, primarily its degree of spherical aberration. There are three basic situations, simulated here: 
In the centre is what you get from a perfectly-corrected lens (or close to it): a disk evenly lit from edge to edge – this is the situation in a large number of modern lenses.
Under- and over-correction of spherical aberration leads to different renderings in the foreground and background. One side of the plane of focus will have what's generally called "smooth" bokeh: as on the left, a bright centre with a gradual fall-off towards the edge. The other side of the plane of focus will have a bright edge with a dim centre, usually termed "harsh" (though not so bad, really).
Slightly under-corrected lenses will typically exhibit smooth bokeh in the background, with the bright-edged type in the foreground. Many of the classic portrait lenses are deliberately designed this way, and it's also what's largely responsible for the very smooth "creamy" look of many older lens designs like the Zeiss Sonnar or Voigtländer Heliar.
Over-corrected spherical aberration will produce the opposite; a smoother bokeh in front of the plane of focus, the harsher one behind. This is often a characteristic of lenses that are both fast and sharp, like many 50mm f/1.4 designs, as well as some macro designs (where sharpness is at a priority).
Distortion: Ideally, all disks would be round and a similar shape, but various other aspects of the lens will stretch them to ovals (astigmatism, field curvature), create small tails (coma), or there may even be mechanical obstructions that cut off edges. All of these tend to be much more obvious as you move towards the edge of the frame.
Color: Chromatic aberrations will colour the disk; lateral chromatic aberrations will create the familiar magenta/green fringing, and longitudinal chromatic aberration will cause a slight colour cast over the entire area of the disk, which will be a different colour in front of the plane of focus than behind.