Q:

Why are functions as a type of relations special? 

I.E. Why are relations where each member of the domain having one and only one mapping to a member in the range so important? 
I was wondering since most of mathematics that I'm familiar with (high school math) focus much on functions. 
Also do upper level mathematics study relations that are not functions just as much? 

A:

When each element of the domain has only one range element associated with it, we have a model of predictable behavior.  In such a case, we can think of the domain element as an input, and the range element as a corresponding output. 
A system in which the outputs can be predicted, based on the inputs, in one that we can study and analyze in many ways, often to very productive ends.  It models cause-and-effect, or the idea of a calculation producing a definite result.  That's why the definition of a function is so useful, and it's what makes functions special, among relations. 

There are other important types of relations that mathematicians look at.  To see some examples, consider the set $\{1,2,3,4\}$, as both domain and (potential) range.  Here are some relations:
$R_1=\{(1,1),(2,2),(3,3),(4,4)\} \\
R_2=\{(1,1),(1,3),(2,2),(2,4),(3,1),(3,3),(4,2),(4,4)\} \\
R_3=\{(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)\} \\
R_4=\{(1,1),(1,2),(1,3),(1,4),(2,2),(2,4),(3,3),(4,4)\}$
Relation $R_1$ pairs elements if and only if they are equal.  It models what we mean by the equal sign: $x=y$.
Relation $R_2$ pairs elements if and only if they have the same "parity", i.e., both even or both odd.  It models what we mean by "congruence,  modulo 2", for which we use the symbol: $x\equiv y\pmod2$.
Relation $R_3$ contains precisely all pairs where the first element is less than the second element.  It models what we mean by "less than": $x<y$.
Relation $R_4$ contains precisely all pairs $(x,y)$ where the number $y/x$ is an integer, i.e., the pairs where $x$ "divides" $y$ (evenly).  It's what we indicate symbolically with: $x|y$
The first two are examples of "equivalence relations", which partition sets into subsets where all elements are somehow "equivalent", for whatever purpose.  Such partitions are used for countless purposes in mathematics, from projections in geometry, to structures such as quotient groups in abstract algebra, to methods of "gluing" shapes together in topology.  Other equivalence relations you may have seen are "similarity" and "congruence" in geometry, often denoted $\simeq$ and $\cong$
The second two relations are examples of "order relations", of different kinds.  Ordering relations are used wherever elements of sets are arranged into orders or hierarchies.  Just like equivalence relations, these are ubiquitous in mathematics.  You can have strict or non-strict orders, e.g., $<$ or $\le$, and usually when someone says "order relation" or "partial order", they mean the non-strict kind.  One important example is the subset relation of set theory: $\subseteq$.

