As of the date of this FAQ (updated 22 Dec 08), Apple has released a total of 17 completely distinct types of iPods, not including color variations and differences in storage capacity.
Full-sized iPods: First (original), Second (Touch Wheel), Third (fully touch sensitive, with 4 separate buttons above the Wheel), Fourth (Black and White), Fourth (Color), and Fifth (with Video), iPod Classic – First (widescreen colour display/two capacities/two colours) and Second (one size/two colours).
iPod touch: First (3 capacities/3.5” display) and Second (rounded back/external volume controls/speaker).
iPod minis (now all discontinued): First (five colors), and Second (four brighter colors).
iPod nanos: First (black or white), and Second (aluminum in six colors), Third (wide screen/5 colours/video playback), and Fourth (9 colours/taller screen/curved profile/accelerometer function).
iPod shuffles: First (white plastic), and Second (aluminum).
There are other, less meaningful distinctions between models as well. Apple’s Identifying iPod Models article lists every distinct model, including capacity differences by release date, and the latest software version for each one.