The available photo printers fall roughly into three categories: dedicated, small format photo printers, general-purpose photo printers and all-in-ones, which can normally handle prints up to A4, and large-format printers for A3 and above, which will probably best suit the semi-professional and fully professional photographer.
Small format printers, then, are typically small inkjet or dye-sublimation printers dedicated to printing photos. Most can produce up to 15 x 10cm prints, though some HP machines can also handle 18 x 13cm prints. Inkjet machines may be three-colour, in the case of HP, or four-colour, like Epson. Canon's SELPHY range uses four strips of dye-sub film.
Dedicated photo printers can produce good quality prints, ideal for the kind of uses to which traditional photos in these sizes would be put. A series of shots from a holiday or a special event are ideal content for this type of printer and you'll typically get natural, well-balanced colour and a good level of detail.
For some years, HP has been trying to curb the use of resolution as the only measure of print quality in a photo. It has a vested interest in this, as it's photo printers typically have a resolution of 600dpi, noticeably lower than Epson's but actually higher than some of Canon's SELPHY machines, which start at 600 x 300dpi. When you look at the images from all three print technologies, though, you can see HP has a point - can you tell the difference?
Just as important as the number of dots is the way in which they can be mixed to produce different shades and the variation in drop size. If you have a large area of the same colour in a given image, printing with fewer, larger dots can give a smoother overall image, as well as being a lot quicker.