Is it possible to plot a graph of any shape? In school, I have learnt to plot simple graphs such as $y=x^2$ followed by $y=x^3$. 
A grade or two later, I learnt to plot other interesting graphs such as $y=1/x$, $y=\ln x$, $y=e^x$.
I have also recently learnt about trigonometric graphs and circle equations.
In the internet, I have seen users posting graphs of different shapes like a heart-shaped graph and a Batman logo-shaped graph.
I am sure there are numerous more graphs that I have yet to see.
Seeing that the graphs can be shaped into shapes like the Batman logo and a heart brings me to my question: Is it possible to plot a graph of any shape regardless of its complexity? Perhaps, shaped into an outline of a person or a landmark? Why or why not?

The answer to your question is yes, you can plot any graph like the one you're interested in. A way to do this (which I'm sure won't be completely satisfying) is to simply have the complete list of all co-ordinate points that you want to graph, and define your mapping m so that $m(x) = y$ whenever $(x,y)$ is a co-ordinate point. (Also note here that $x$ might map to more than one $y$ - for example in your heart shaped example). 
What it seems like you'd like however is a 'nice looking' map like $y = e^x-4x+1$, or something like that. In order to do something like this, many methods exist. I would suggest looking up interpolation for a start.
