Why does (1/3) mod 3016 = 2011? So I am taking a class where we are working on a cryptography section. Basically, the course says that:
$$\frac 1 3 \mod(3016) = 2011$$ 
or when run through Python - modified with SciPi:
$$\frac 1 3 \,\%\, 3016 = 2011$$    

I don't understand how or why this happens. Does anyone know why the modulo division would produce this result?
What you're calling $\dfrac 13$ is just the multiplicative inverse of $3$ mod $3016$.  You can easily verify that $2011$ is the multiplicative inverse of $3$.  $$3\cdot 2011 = 6033 = (2\cdot 3016)+1$$
No one has mentioned that for any fixed N, {a | gcd(a,N)=1} forms a group. Closure under multiplication is easy to see with elementary manipulations.
The inverses usually are explained using the Euclidean Algorithm. At the very least if you want to understand inverses in modular systems, this desire could sustain you while you read/hear an explanation of the Euclidean Algorithm.