For starters, it's worth pointing out that there are two general categories of images; vector [e.g. SVG, WMF, Adobe Illustrator and Corel Draw Graphics] and bitmap (also called raster) images [e.g. Bitmap, JPEG and PNG Images].
Vector images are formed from a series of mathematical equations and/or calculations. Bitmap images, on the other hand, are made up of individual dots (pixels) each corresponding to a particular feature on the object the image is taken of.
If it should happen that you want to resize an image, the first thing to consider is if it is a bitmap or vector image. By virtue of the fact that vector images are obtained from calculations, they can be perfectly resized without losing any detail. The case is different for bitmap images. Since each pixel is independent of the other, when you desire to resize it, you are simply increasing or decreasing the source to target pixel ratio.
So in order to double the size of a vector image, simply multiply the target dimensions by two and everything comes out all right. If you should apply the same effect on a bitmap, you are actually increasing each source pixel to cover four pixels (two rows of two horizontal pixels).
Of course, by applying interpolation and filtering, the computer can "smooth" out the edges of the target pixels so they seem to blend into each other and give the appearance of a reasonably resized image but this output is never the same as resizing a vector image; vector images resize perfectly.
You also mentioned DPI in your question. DPI is essentially the number of pixels that correspond to an inch when the image is printed not when it is viewed on a screen. Therefore by increasing the DPI of the image, you do not increase the size of the image on the screen. You only increase the quality of print [which needless to say depends on the maximum resolution of the printer].
If you really desire to resize the image and the image is a bitmap, as a rule of thumb, do not increase the size beyond 200% of the original image's size else you'll lose the quality.
You can see this answer for code to resize bitmap images.
To see a sample vector image, go to this link.
Note Try zooming in and out of the image to see how well it resizes.
A typical bitmap are the StackOverflow sprites. They do not keep their quality resized.
Further Reading