DPI stands for dots per inch and it determines clarity and crispness to a display. This is determined by the size of the screen and its resolution. The higher the DPI the less the individual pixels can be discerned on the display. This is also dependent on the distance that the screen is being viewed from. At a long enough distance the human eye will not be able to discern individual pixels regardless of DPI and will look the same.
Most monitors range around 100-200 DPI(depending on the screen size) while smartphone screens are more condensed and range from 300-600 DPI. Ideally you want higher DPI for best clarity and picture quality crispness, keep in mind that in regards to computer monitors and computer systems higher resolution/DPI requires more graphical horsepower and so will require a higher end video card to power the system.
In regards to smartphones higher DPI screens will have higher resolutions and will generally consume more battery life and graphical horsepower to power the screen.