Fuzz, Distortion and Overdrive
I have grouped these three together since they all do a similar thing though there are subtle differences.
The Fuzz pedal was one of the earliest pedals produced. The sound was meant to reproduce the sound of an amplifier breaking up though as players wanted more gain and sustain. Fuzz pedals tend to have a lot of gain (distortion) so are popular for lead playing. They can sometimes sound a little out of control on low notes. A good fuzz pedal will respond to pick-up changes. Listen to the examples recorded on the front pick-up and then the front pick-up with the tone control down.
Distortion pedals gives a harder edged tone by adding odd harmonics to distort the waveform. Distortion pedals are more popular in heavier styles of music where a harder edged distortion is required.
Overdrive is softer distortion, designed to sound more like an amplifier (and in particular a Valve amplifier) distorting. This is more popular in classic rock and blues styles.
There are many variations on the three basic types. You can of course use your amplifier's overdrive. Most modern guitar amps offer enough distortion not to require an separate pedal.
There are also hybrid overdrive / distortion pedals and pedals that use a valve in circuit to produce a sound more like an valve amp. A Treble Booster boosts the treble frequencies to overdrive an amplifier. Bass boosters and full range boosters have also been produced. Though these are associated more with the 60s they have become popular again with modern players looking for a more vintage tone.