Phi is the basis for the golden ratio, which is the sum of the two numbers before it. This can form a neat spiral, as shown: (place an image that you find that's the golden ratio spiral)
If you break a stick into two, you'll have two sticks now, a and b. if the larger segment, which we'll say is a, is the same ratio to b as to the whole thing before the stick was broken, the ratio will be phi.
Phi can be seen in various designs, such as some types of architecture, some shells, and lots of other designs.