It is absolutely vital to use the correct ink for your printer if you want the best results. Some inkjet printers may be able to use third-party cartridges or refillable ink. However, I strongly advise against using them, in particular Epson printers which have a totally unique way (piezo) of propelling the ink onto the paper. One exception to this is Lyson inks which have been specifically designed for use with some Epson printers, but only after you have completely flushed out any Epson ink first. The advantage of using Lyson inks is to prevent the colours quickly fading, a notorious failing of all inkjet inks. Remember that you can always reprint the picture if it fades, without resorting to using special inks.
However, a new development in Epson inks used with the new Photo 870 and Photo 1270 printers should equal the colour fastness of the Lyson inks. They could even possibly better the longevity of Lyson inks, there are some many claims flying around. The latest Epson cartridges may even use Lyson inks, as I know Epson has tested them. To achieve this colour fastness the use of the approved paper, either Epson or Lyson, is just as important as the ink itself.
From Sep 2001, Epson are now using coloured pigment inks in the new C70 an C80 printers to improve colour longevity, similar to the Photo 1290.
Some printers, such as the Epson Photo range, use two additional cells, a lighter Magenta and a lighter Cyan, to compliment the usual Cyan, Magenta and Yellow. This provides a more even ink coverage in lighter areas, such as with sky or skin tones, and allows more subtle tones of colour to be reproduced giving a more photographic quality. All printers have a separate black cartridge and a few (but not Epson) have a lighter yellow as well, although this is so close to white it would only be of use on dark paper which you wouldn't normally use for photographs as you could not create white anyway using the colours available.
Many friends at Photocraft camera club have recommended to me the MX2 web-site for cut price consumables, including the full range of genuine Epson inks.