Why does the empty set matter? I'm trying to understand why the empty set is a useful tool for mathematicians. Are there any nontrivial theorems that could only be achieved by the existence of the empty set, or is the recognition of the empty set just a conventional standard that mathematicians have adopted?
 A: The same reason it's a useful tool in everyday life.
There are lots of questions that ask for sets as answers, like...


*

*Which crew members are still aboard the airplane?

*What dishes are in the sink and need to be put away?

*What problems does this piece of software have that must be fixed before we can release it?

*What songs have you practiced on the piano lately?


And sometimes the answer to one of these questions is, "There aren't any." That's the empty set.
A: Often, including the possibility of a set being empty in your proof makes it shorter.  It allows you to avoid writing down multiple cases.  For example, if $E$ is specified to be finite set of nonzero real numbers, one may want to do something with their product
$$\prod\limits_{x \in E} x$$
This will still make perfect sense if $E$ is empty, if one follows the convention that $\prod\limits_{x \in \emptyset} x = 1$.
In field theory, one may work with a purely transcendental extension of fields $F \subseteq F(S)$, where $S$ is a set of algebraically independent elements over $F$ living in some large field $\Omega$ containing $F$.  If $S$ is finite, say with $n$ elements, then $F(S)$ is just the field of rational functions $F(X_1, ... , X_n)$ in $n$ indeterminates.  But one may want to keep the possibility in mind that $F(S)$ is equal to $F$, so we would allow $S$ to be the empty set and define $F(\emptyset) = F$.  
Also in many proofs by induction on the size of a given set $A$, the proof works fine if you do the base case where $|A| = 0$ ($|A|$ being the number of elements in the set $A$), and it's an easier base case than $|A| = 1$.  If you ever happen to read Bourbaki, you will see this done a lot.  
Many theorems are perfectly valid when some set in the statement of the theorem is allowed to be empty.  But the question you might ask is why we would care about allowing something like that in the first place?  In long mathematical treatises with complicated logical interdependencies between different theorems, one can often shorten a proof by citing a previous theorem in a trivial way, taking some set in the statement of that theorem to be empty.  Again, any treatise by Bourbaki does this a lot.   
Overall, I don't think the empty set is really necessary to state or prove most mathematical results.  But without it, proofs and statements of many useful theorems would be longer and broken into more cases.
