What does it mean when $|z| = 2$ is the curve in a contour integral? 
B) Evaluate:
$$\oint_{|z|=2} \tan{z}\,dz$$
Specifically looking at B on the image. What is meant by $|z|=2$?

$|z| = 2$ is the circle of radius $2$ centered at the origin. Typically, when one writes an integral like
\begin{align}
\oint_{|z| = r} f(z) \, dz,
\end{align}
what is meant is that we have to consider the path $\gamma: [0,2\pi] \to \Bbb{C}$ given as $\gamma(t) := re^{it}$ (so the orientation of the path is counter clockwise). And then, really we're supposed to compute a line integral:
\begin{align}
\int_{\gamma} f(z) \, dz &= \int_0^{2\pi} f(\gamma(t)) \cdot \gamma'(t) \, dt \\
&= \int_0^{2\pi} f(r e^{it}) \cdot ir e^{it} \, dt.
\end{align}
