I am reading Gunnar Carlsson and Afra Zomorodian's "The theory of multidimensional persistence" and I'm stuck on what an n-graded vector space V is. Their definition seems to be that V can be written as $$\bigoplus_i k(v_i)$$ for some family of vectors $$v_i\in\mathbb{N}^n$$. In particular, k is a field that is $$\mathbb{N}^n$$ graded as follows: $$k_0\cong k$$ and $$k_v={0}$$ for all other v. Thus $$k=\bigoplus_v k_v$$. The notation $$k(v_i)$$ means to shift the grading to $$k(v_i)=\bigoplus_v k_{v-v_i}$$.
What I'm missing is that I don't understand at all what this grading means. What does it mean that $$k(1,0)$$'s generator lies at $$(1,0)$$? What does the vector space $$k(1,0)\bigoplus k(2,3)$$ look like?
My second question is why $$\rho(M)$$ is an n-graded vector space. Here, M is a finitely generated $$\mathbb{N}^n$$-graded $$k[x_1,...,x_n]$$ module and $$\rho(m)=k\bigotimes_{k[x_1,...,x_n]}M$$ where k is given a $$k[x_1,...,x_n]$$ module structure s.t. $$x_i$$ acting on k is merely the zero map for all i.