Understanding of graded algebra I am recently learning from Loring W. Tu's An Introduction to Manifolds the concept graded algebra, which is used for introducing exterior algebra. I don't understand the following definition:

An algebra $A$ over a field $K$ is said to be graded if it can be written as a direct sum $A=\bigoplus_{k=0}^{\infty}A^k$ of vector spaces over $K$ such that the multiplication map sends $A^k\times A^l$ to $A^{k+l}$. 

Here are the questions:


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*What does $k$ in $A^k$ mean? Is it a superscript or the power of $A$? (If it is the power, what does $A^0$ mean?)

*Since I don't know much but some very basic knowledge in abstract algebra, I am trying to understand the concept with some simple examples. (I don't quite understand the wiki article of graded algebra). Is there any example as simple as the linear space ${\mathbb R}^n$ for the graded algebra? And what does it look like concretely?


To add to Qiaochu's answer, often one can "explain" the existence of a grading on a mathematical object by giving a representation of the "circle group" (analytically, $S^1$, algebraically, the multiplicative group). The idea is that a representation $V$ of the circle group (here I'll use the analytic language) canonically decomposes into Fourier modes: the irreducible representations of $S^1$ are parametrized by the characters $\chi_k: t \mapsto t^k, S^1 \to \mathbb{C}^*$, and any nice representation $V$ splits as 
$$V = \bigoplus_{k \in \mathbb{Z}} V[k]$$
where $V[k]$ denotes the subspace of vectors $v \in V$ such that $S^1$ acts on $v$ via the one-dimensional character $\chi_k$. Conversely, given a graded vector space $ V  = \bigoplus_{k \in \mathbb{Z}} V_k$, one defines a circle action by letting $S^1$ act on the $k$th piece by $\chi_k$. 
