I've had my HP printer for several years now and I've always been very happy with the quality of prints it produces. I use it to make all the prints (ACEO / 5x7 inch / 8x10 inch) in my shop. The printer takes two cartridges - one tri-colour (cyan, magenta and yellow) and a photo cartridge (black, light magenta and light cyan) so I get a good range of colours in the prints. I print on HP's Premium Plus range of photo paper to get the best results - it also means that if kept out of direct sunlight the prints will resist fading for over 100 years. Photos which I printed on Premium Plus paper at least 5 years ago look like new, whereas photos I printed on cheaper paper look faded after only a short time. (I've not been around long enough to test the 100 years yet!!)
The HP inks I use are dye based which, although good for my art prints, has one downside for printing scrapbooking papers - the ink runs if you use PVA glue or glitter glue on them. Dye based inks are water soluble coloured liquids which are absorbed by the surface they are printed on.
Pigment based inks - like those made by Epson - are made from ink particles encapsulated in a resin coating which sit on the paper surface and are not absorbed into it. They produce bright colours, will last up to 200 years and are water resistant - so are great for printing scrapbooking papers. I have access to an Epson printer (which produces great prints!) and I'm using it to produce the papers for my cardmaking kits.
One downside of the pigment inks is that they're not so good for printing on fabric that will be washed often as the ink sits on the surface instead of soaking into the fabric like dye based ink. I'm experimenting with some fabric printing at the moment, so I'll have to see how the two printers compare.