A Distortion pedal functions by ... well, by "distorting" the original sound wave. Some characteristics of a sound wave is a certain shape, a certain amplitude, and a certain frequency. A distortion pedal will distort some or all of these characteristics by electronic manipulation to achieve the desired distorted tone.
Typically distorted sound is "dirtier" than the original "clean" sound. If you're playing "edgy music" ... rock, hard rock, grunge, metal, things like that ... you're probably seeking a distorted sound.
While current distortion pedals manipulate the tone through solid-state electronics (most often -- some pedals use tubes, but these are most effective in "overdrive" pedals), musicians in the pre-pedal world often used damaged equipment -- amplifiers with bad or missing tubes, speakers cut with slits or tears, etc -- to achieve a distorted sound!
A distortion pedal and a "fuzz" box are basically the same thing -- both distort the original clean signal. The difference between the two is the degree and character of the distortion.
You probably do not need both a fuzz and a distortion pedal. But ... depending on the style of music you're seeking -- you'll find that one works better for you than the other. The only way to determine which is best for you is to take some time at your local music store and try several pedals.
Distorted guitar offers a thicker and heavier "base" -- a single guitar can provide a solid backing for a tune. Distortion also leads to greater sustain and can feedback in a pleasing manner. Distortion is most commonly used for solo guitar ... but is also effective when using power chords. However, fully voiced chords quite often lose their punch when distorted. Distortion can also be effective for bass guitar ... although you will generally use a pedal specifically designed for a bass.