You have 2 main choices:
Hewlett Packard & Lexmark etc have the print heads as part of the cartridge - this means that every time you change the cartridge, you're also putting fresh heads in - there's less problems with having to clean the heads, but the cartridges are a lot more expensive. They're a good option if you don't plan on printing much.
Epson & Canon etc have the print heads as part of the printer itself - cartridges are therefore much much cheaper, and there are some very good compatible cartridges around too - but you may have to clean the heads from time to time. Because of the cheaper cartridges, though, it's a great choice if you're likely to print a lot.
In terms of print quality, all the big brands turn out fairly good prints these days.
One other thing to consider - you may find it works out better to have your photographs printed at Walmart or somewhere similar, or one of the many online processors instead of printing them yourself - by the time you've bought all the photo paper and inks, it works out about the same price.
(Oh, btw, I've got a collection of Epson's myself, but mainly upload to online labs)