I know that it has something to do with Elliptic Curves in the Complex Plane but I don't have an intuitive sense as to what its there for.
I understand it is defined in terms of multiple Einstein series and is a modular form of weight $0$ but outside of that, I am really confused about this function.
If someone can explain the j-invariant without having to go over the entire subject of Elliptic Curves that would be greatly appreciated.
Let me give a crude answer not in the language of complex analysis but in purely down-to-earth algebraic terms.
If you have a plane curve $E$ of genus one given by an equation $Y^2=X^3+aX+b$, there’s a rational expression $j(a,b)\in\Bbb Q(a,b)$ with this property: if you have another curve $E'$ given as $Y^2=X^3+a'X+b'$, then over an algebraic closure of the field containing the coefficients, $E$ and $E'$ are isomorphic if and only if $j(a,b)=j(a',b')$.
The above is imprecise and sloppy, but good enough for an appreciation of what $j$ does: it classifies the elliptic curve $E$, but unfortunately only over an algebraically closed field. Two curves with $\Bbb Q$-coefficients can have the same $j$-invariant, but not be isomorphic until you make a finite extension of $\Bbb Q$.