What is Microtonality? When and why is it appropriate to use these symbols? Why do they even exists?
Microtonal music is music that uses microtones -- any interval that is smaller than the usual half-steps (each 1/12th of an octave) that are found in traditional western music. It can actually be rather tricky to define exactly, because many definitions exclude scales that are audibly indistinguishable (or nearly so) to western music (e.g. quarter-comma meantone uses pure/just thirds that are somewhat flat compared to 12TET, but still creates a 12-note scale that sounds very similar to 12TET, and would not be considered truly microtonal).
Another common generic term that can be used is xenharmonic, which means strange, or foreign sounding. There is a good website that deals with all manner of microtonal and xenharmonic musical topics called the xenharmonic wiki. Specifically, they have a good introduction that answers the question "what is microtonal music", including some of the difficulties in defining it, and some of the alternative names that are used.
Broadly speaking, you can think of such music as coming from either a non-western source (many of the world's musical cultures do not use the 12-note system of western music), or else coming from a western, but non-traditional source (often referred to as "experimental" music).
As an example of the first, the Arab tone system uses a 24-tone equal system, from which a 7 note scale is then chosen . This leads to notes that are a quartertone apart. Some_Guy has mentioned Blues, rooted in African-American music, which uses "blue notes" that are flatter than normal, but by less than a half step.
Since western musical notation was originally designed to handle western music, it does not necessarily have symbols to handle notes outside this system. Thus symbols such as the ones you ask about (half and three-quarter sharps) were developed to fill this need.