This question actually is a little complex, but I'll try to explain without getting flamed for being too general...
Basic 4-colour printing: CYMK
Basically, all printers operate on the same principle of 4-colour printing to create the entire spectrum of colours. These colours are cyan, yellow, magenta and black. These printers are basic, but nowadays even these basic printers produce excellent photos which you'd be hard pressed to tell from photo lab prints. Given a calibrated workflow, I'd have no qualms or misgivings to use even a 4-ink photo printer. They are that good nowadays...
5-color:
The so-called 5-colour printers are actually more like 5 ink printers, because they are actually 4-ink printers with additional (larger) pigment black cartridge that is used for printing black text. So you get sharper black text with the extra pigment black. In terms of photo printing, they're using the same 4 colours, so there's no difference in the quality.
6-colour:
The basic 4-colour printers can produce a wide range of colour, but there's the issue of the in-between colours. When a image requires a shade of cyan that is 50% the intensity of the regular cyan ink, how can the printer produce such a shade? It cannot add water to dilute the intensity :bsmilie: , so the only way is to dither the pattern so that the paper base shows through slightly to reduce the intensity to fool the eye. This also means that the 4-ink printers cannot produce some shades and gradation smoothly. So the manufacturers added 2 additional inks - the photo inks. The photo cyan and photo magenta are lighter versions of the basic cyan and magenta inks, of say 1/8th the density of the regular inks. So when a particlar shade of say 1/2th density of the regular cyan ink is required, the printers will spray the photo cyan inks 4 times to build the density to the required 1/2th density of the regular ink. Hope I didn't lose you... it's pretty simple really. This is also why the photo inks will be used up the most quickly amongst all the inks, and why the individual ink tanks system is really good in this aspect because you can replace only the ink that runs out.
7,8,9,10 inks etc
Any ink more than 6-ink actually serves to expand the colour gamut. The 4-ink and 6-ink system does not really cover some of the areas, so manufacturers added red, blue and green ink etc to help to push the areas of colour reproduction even further. If you compare a print made with such a printer against a 4 or 6 ink printer side-by-side, you will be able to find some discrete differences in certain areas, but for most consumers the differences are not that apparant.
The other reason is that the colours added such as blue, green and red are difficult to reproduce, since they require the printer to mix the colours. Pure chroma red, green, blue don't exist in CYMK technology, and the resulting colours may have a slight dirty hue instead of a pure hue (eg. green is made by mixing yellow and blue). So having pure chroma inks such as additional red, green and blue helps to provide pure colours and expand the colour gamut .
Finally, some of the extra colours are shades of grey. These come in very useful to expand the grayscale printing capability of printers, giving you very good monochrome printing. Else the printers usually mix minute amounts of colour ink with black to achieve smoother tones in monochrome printing, but at the expense of colour cast. So having multiple black/grey inks actually help you achieve truer monochrome prints.
Phew! :sweat: :sweat: :sweat: