I don't think any conversion is relevant: Dots and Pixels are different things -- it's like comparing apples with pears. The size and number of pixels per inch of your screen is determined by the screen manufacture. The number of dots per inch in an image is determined by (a) how the hardware and software have recorded the image and, (b) the resolution of the printer that's printing it, or the screen that's displaying it.
Perhaps if you could say why you're interested in converting one to the other, we could answer your query better.
horiz5, I have been asked by a friend who has just bought a new digital camera, & he is trying to work out if his printer is good enough to print a picture of equal resolution to the camera. stlucia, thank you, I already thought that it was not really related but it answers my question.
Okay, so let's say he gets a 3 megapixel camera. That means there's 3 mega pixels in the whole photo which, if it's going to be printed at 6" x 4", means there's 125,000 pixels of information for every square inch (3 million divided by 6 divided by 4), which is 353 pixels per linear inch (square root of 125,000). Therefore, in theory, a printer that can print 353 dots per inch will be optimum for that camera and that size of print.
That calculation can be extended for any size of print, but I think it will reassure you that many modern printers can do full justice to the output from a digital camera.
Pixels per square inch has no real meaning, unless you take into account the size of the picture.
A 3Mpixel camera image has 3million pixels in it - whether it's viewed at 1"x 1" or 10" x 10" - clearly the pixels per square inch are 100x times different in the one view than the other.
stlucia's comments should tell you that almost any modern printer will do the photos justice - though there are other important parameters in a printer than just dpi - such as number of colours, and droplet size.
If an image is printed at 1 dot for each pixel, then the quality will not be at an optimum. The reason being that each pixel will then have to represented by only one of the printers 4 or 6 colours. It is better to print at higher dots per inch.
For example, most of my images are at a resolution of 300 pixels per inch. I print at over 1200 dots per inch. Therefore each pixel is represented by 16 dots. This gives much smoother shades of colour.
For more detail, search for digital photography printing tips in Google. This site explains the relationship/difference between ppi and dpi quite well.