Megapixels and file size have virtually no relation to each other. A 10-megapixel photo might "weigh" less than a megabyte on your hard drive. Or it might "weigh" as much as 6 megabytes. The file size depends on several factors, including the number of megapixels, the file format you're using (such as JPEG or RAW), and the amount of file compression used to save the photo, which is sometimes referred to as the quality setting.
DPI Measures the Density of the Pixels
Finally, there's the thing that Sue actually asked about: dots per inch, or dpi. Dpi has no inherent value of its own when describing the size of a photo. The only thing that dpi does is help you to understand how large a photo can be printed or displayed, and--here's the key thing--it refers to the display medium, not to the photo itself.
What am I talking about? Suppose you take the 10-megapixel photo I mentioned at the beginning of this article and display it on a computer screen. Computer screens tend to have a resolution of around 72 dpi, which means the screen has about 72 pixels per linear inch. If you show the photo at its "full size" (so every pixel in the photo uses a pixel on the screen) then you'd divide 3872 by 72 and find that the photo would be about 53 inches across. But send that same photo to a 300-dpi inkjet printer, and you would expect that you could make a high-quality print that's about 12 inches across (3872/300).
So what does all this teach us? A few things:
Megapixels are a general guide to the size of the photo, as measured by the number of pixels it includes. Megapixels is a strong indicator of quality in the sense that it helps you know how large the photo can be safely printed--more megapixels equals a larger print--but it doesn't really tell you anything about the quality of the camera's sensor or its lens. And other factors, like a high level of JPEG compression, can affect the quality as well.
Megabytes tells you how much space a photo takes up on your hard drive, and has nothing to do with your camera's megapixels. The same photo, saved at different JPEG quality levels, will yield wildly different file sizes in megabytes.
Dots per inch (dpi) is just plain meaningless most of the time. You can use this number, along with the photo's resolution, to find how large it can be printed or displayed on a particular device. But to be useful, you need to know the dpi of the device in question--for example, most inkjet printers give good results at no more than about 300 dpi.