Definition of a Grothendieck ring I've been looking at some definitions of Grothendieck rings. However I've not found a good definition that I've understood. Any recommendations?
  I'm referring to the definition in tensor categories, more specifically I've discovered there are some structure coefficients in a Grothendieck Ring, I understand the mathematics from a physics perspective. I learned this first from Physics papers, but I want to understand more deeply the mathematical structure.
http://www.math.sunysb.edu/~kirillov/tensor/tensor.html is the set of lecture notes I'm referring to, and I'd love if someone could help me tag this post more effectively. 
 A: Let $\mathrm{Var}_k$ denote the category of varieties over a field $k$. Then $K_0(\mathrm{Var}_k)$ is the free abelian group generated by symbols $[X]$ for $X\in \mathrm{Var}_k$, subject to the relations:
(i) $[X]=[Y]$ if $X \cong Y$;
(ii) $[X]=[X\setminus Y]+[Y]$ if $Y$ is a closed subscheme of $X$ (the so called scissor relation).
We can define a multiplication on $K_0(\mathrm{Var}_k)$ by $[X]\cdot [Y] := [X \times_k Y]$. The resulting ring is called the Grothendieck ring (of varieties).
A: I'll expand my comments into an answer. Since I'm not quite sure what parts bother you,
I'll assume it's everything! The Grothendieck construction is actually
a family of related constructions, which is brilliant in its simplicity.
Whenever, you have a collection of things (e.g. finite sets) that can split
into parts, you can force it  be an abelian group by requiring that the sum of parts
correspond to addition in the group. If your things have more structure, then the Grothendieck group can be expected  to inherit this as well.
To get closer to what you seem to be interested in, suppose that $G$ is a discrete or Lie group,  and $C$ is the category
of finite dimensional complex representations (as a Lie group). Then $C$ is a good example
of a tensor category: It's abelian, so we can speak of exact sequences, there are also tensor products (usual product with $g(v\otimes w) =gv\otimes gw$), and various compatibilities hold.
Given $\rho:G\to V$, one can attach a character $\chi_V= g\mapsto trace(\rho(g))$
which sometimes determines $V$  when $G$ is compact, but not in general.
Nevertheless,  the standard relations


*

*$\chi_V = \chi_{U}+\chi_{W}$ when $V$ is extension of $W$ by $U$

*$\chi_{V\otimes W} = \chi_{V}\chi_{W}$


always hold, which makes this notion quite useful. 
Now suppose that $C$ is a more general tensor category. What would be the analogue of the ring of characters on $G$? It would be the Grothendieck ring, generated by symbols $\chi_V$
where we simply impose the above relations.
