That depends entirely on your needs. Frankly, anything you can get on sale at Walmart for $40 these days will run circles around the $20K Tektronix proofing printer/RIP setup I was using in the early '90s, so if your needs are truly casual (occasional printing only, "good enough" really is good enough, small print sizes) then there's no need to spend a lot of time soul-searching -- just get whatever seems to have the consumables (ink and paper) handy.
If you want to get a bit more serious, then the basics quickly become these:
Individual ink tanks/cartridges. There are few things more depressing than throwing out cartridge after cartridge because, say, the yellow's done but the other colours are still nearly full.
More than four ink colours. Having lighter (or more dilute) versions of black and one or more other colours means that your prints will have smoother apparent tonality and will be far less "grainy" when seen up close.
There are a number of reasonably-priced letter-sized printers that meet these minimum standards, notably in the Epson "Artisan" and Canon "PIXMA" lines. (You'll probably find that the choice at most levels, at least with inkjets, will come down to Epson versus Canon. There are other players in the game, like HP, Lexmark, Brother, and so on, but they concentrate on business and general-purpose needs rather than on photography.) Printers in this range limit you in two ways, though: the maximum size of the print, and the kinds of paper that the printer can handle well. If you can live within those limits, then you are well-compensated by the relatively low price of admission.
The next step up is to the larger-format offerings: the 13" printer class (the standard sheet size is 13" by 19"). They're big, and can sometimes take up a lot more room than you were hoping to spare them (remember that you need space for paper in and paper out when they're operating). Not only does that mean you can print larger pictures, you get:
Eight or more ink colours (in current models). How these colours are arranged varies by manufacturer and model, but there is usually at least a lighter cyan and a lighter magenta in addition to the lighter black, again giving the print smoother tonality. There may be one or more intermediate colours as well that are hard to achieve by mixing CMYK values. There may be a "light light black" as well -- which makes a huge difference when you're printing B&W work (even with a toned effect).
Two blacks. Not two different "shades" of black, but two different formulations, one for glossy prints and one for matte prints. The printer may automatically swap them, or you may have to make the change yourself, but you usually do get the option to use the right black for the right paper at this level, and it does make a difference.
Better media handling. There are usually a number of feed options allowing you to use thicker or more highly-textured materials to print on. Often (but not always) there's also an option to use roll paper for panoramic or banner-format printing.
Both Canon and Epson offer 13" models that can usually be found on sale somewhere at about the $300 range. (Currently, that would be the Canon PIXMA Pro 9000 MkII and the Epson Stylus Photo R1900.) This is probably the sweet spot in bang-for-the-buck terms. You get great-quality output, but you don't get all of the features that a professional would depend on for day in, day out work. Oh, and your prints will probably only last 50-75 years rather than 150+, if that's not a huge problem. You trade off a few minor workflow annoyances to save hundreds of dollars at this level, and neither printer is fundamentally better than the other. You are sort of locking yourself into that brand's consumables (inks and paper†) when you make the choice, but both are widely-enough available that it shouldn't be a concern.
Beyond this level, you are leaving the hobbyist scene behind. The pro-level 13"-class printers start to get expensive, but you're paying for things that a pro would need (workflow, media handling, archival assurances that make print sales more attractive from a consumer rights point of view, etc.). You do get a slight bump in quality, but you're entering the realm of diminishing returns if you're not creating prints for sale. And after that, you're looking at the wide-carriage class (17" or wider), that can offer things like inline calibration in addition to printing huge prints. But if you aren't planning to print 16x20s (or larger) on a regular basis, it's much more cost-effective to have your occasional large print done by somebody else.
† You can print on third-party media, but it often means a lot of experimenting, and may mean creating or obtaining a custom paper profile. The manufacturer's own brands generally "just work", since they're developed in concert with the inks and have the absorbancy, holdback and bleed best suited to those inks.