This question has come up often enough to warrant putting its own page:
What resolution images do you render to get such clear and detailed prints?

1) Print

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:

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).

2) Electronic Images

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.

3) Notes

A few complications might bear noting.