I need clarification on the relationship between DPI, the pixel dimensions and the inch/cm dimensions of an image.

For example, why is it if I increase the DPI of an image but keep the dimensions the same it will print no bigger but look gigantic on my screen?

Yet sometimes if I want to decrease the quality of a hi-res image I can lower the DPI and shrink the image, or lower the dimensions and keep the original DPI?

In other words, I posses only basic knowledge of photoshop and am thoroughly confused about how DPI and dimensions relate to one another and determine the size of an image.