A: In the 'Good Old Days' continuous tone originals had to be 'screened' to convert them into small dots that could be printed. This was achieved by exposing the original photo through a finely-ruled screen onto photo-sensitive film. The most popular screen 'pitch' was 150 lines per inch (LPI).
When electronic drum-scanning replaced cumbersome photo-optical methods, it was found that generally, photos had to be scanned at a higher pitch than the screen to reproduce fine detail. Doubling the screen ruling - i.e. 300 DPI - was used as a good rule of thumb: it's now more or less universally accepted as a standard, even though 200 DPI can be perfectly acceptable.
Now, in the digital age, many print technologies are designed and built around the '300 DPI standard'.
On it's own, '300 DPI' means nothing: combined with the physical dimensions of a required reproduction, it tells us how much information - the number of pixels - is required to ensure a good print.
For example, let's say we want to produce some 5x7 inch postcards. We need (5x300) x (7x300) pixels: 1500 x 2100 pixels, something that most digital cameras can achieve. But if we crop our photo heavily, we can easily end up with a lot less pixels than we need, and as we can't invent detail that was never there, we'll end up with a fuzzy, blurry print.