This question has come up often enough to warrant putting its own
What resolution images do you render to get such clear and detailed prints?
The problem with this question is that there are several different kinds of resolution at work when printing an image, that have to match each other and that are easily confused -- especially since even many "professional" places on the web keep mixing them up and using them in the wrong context. Here they are, with a little background:
Note, that most modern inkjet-type ("giclee") printers have only four different pigments: cyan, magenta, yellow and black (often abbreviated to CMYK. There exist eight- and even sixteen-color print systems, but they're rare and usually reserved to a fairly upscale market). This means that a 300dpi print process can generally not print 300 pixels per inch, if you want your pixels to have any more than 4 or 8 different shades. Which gets me to
256 is 16x16; which means that faithful representation of 256 different shades of, say, magenta requires putting 16x16 droplets of ink on the paper for each pixel that you want to print, so as to fill the pixel more or less with magenta pigment. Most modern printers should be able to fine-weave, i.e. put one drop of each pigment on any given spot on the paper; which means that each "droplet" can have one of 8 different shades (C, M, Y, C+M, C+Y, M+Y, white and black) and thus that 16 droplets of ink in each direction should be sufficient to reproduce any color you'd like (except for the gamut issues mentioned here).
From this it follows that ppi (a property of your print) and dpi (a description of your printer) are not in the least interchangeable and their frequent mix-ups even by professionals is unfortunate. For almost all print-jobs, 200 or 250 ppi should be sufficient; while 200dpi is definitely not enough. Even marginal black-and-white printers deliver 300dpi these days, and for a professinal print job, I'd never go with anything less than 1200dpi. Remember: 1200dpi only gives you 6x6=36 different shades per pigment for a 200ppi print - which I do not consider terribly excessive.
While a printer dpi can never be "too high" (except for questions of cost), paper can have a resolution that is "too high". In general, you will want a paper resolution that matches your ppi - this will assure that the multiple dots of pigment that your printer generates per pixel will properly bleed into each other and create appropriately blended shades of color. If you paper resolves too high, you can sometimes see the individual ink-droplets used to compose the pixels, which looks a little like half-toning in the newspaper and is usually not what you want.
On the other hand, you don't want your paper of such low quality that the pixels of your image start blurring into each other - that would just waste computational effort.
At this point I can make the first statement about the resolution at which I print: I find that 200ppi are practically almost always visually indistinguishable from 250 and thus 200ppi is a fine print resolution for all but some fairly specialized cases. I happen to work with a print shop that prefers 250ppi, so that's what I use often - but that is chosen because I know this is "good enough". In almost all cases, the paper should match the print resolution (in ppi) and the printer should have as many dpi as is necessary to get the required color resolution into the available spatial resolution.
How much color resolution is needed? Well, for my fractal work I prefer fairly high values, since sometimes much effect hinges on fairly subtle color differences. Which means going with pretty high dpi on the printer side. But it also means that I can get away with paper that is geared towards things other than spatial tightness. I personally like Metallic Lightjet C Prints, which are preferred by car-shooters and similar glam-print (= high-budget) folks. Can be had at many high-end labs, for example Nardulli here in Hollywood. If, on the other hand, your images rely on high contrast you may not need such a fine color-granularity and might instead prefer to go for sharper spatial expression (or slower film).
At this point I still haven't answered the question "at which resolution I render" because it should now have become clear that a rendered image in any kind of pixelated electronic form (JPG, PNG, TIFF you name it) does not have a resolution. What it has instead is an extent, measured in pixels on each side.
For example an image might be 1600 pixels wide and 1200 pixels high - in which case it would be as large as my monitor. Which happens to have a 21 inch diagonal. If it had a 22 inch diagonal, the same number of pixels would be displayed on a larger area, resulting in a different resolution. Printed at 200 ppi, this image would be 8x6 inches - printed at 400ppi it would be 4x6 inches. Conversely: printed at 16x12 inches in size, it would have 100ppi resolution.
The size of a pixel (or the number of pixels per inch) only ever enter when the image is to be output to some device - be that a monitor, projector, printer etc. It is thus a property of the output device, not a property of the image.
In normal shoddy everyday language it is unfortunately common to refer to an extent like "2048x1536" as "resolution". Or, worse yet, "3 Megapixel resolution". Which is OK, as long as it is understood that something like accurate printing requires refinement of these terms.