In Atiyah i recently read the definition of a graded ring as a ring that can be written as $R=\displaystyle \bigoplus_{i \geq 0}^{\infty}R_i$ where each $R_i$ is an abelian subgroup of $R$ (with the condition that $R_iR_j \subseteq R_{i+j}$.
However I find this confusing, because in other textbooks I read definitions with $R=\displaystyle \bigoplus_{i= -\infty}^{\infty}R_i$. Why does one definition use negative index but the other doesnt? I also stumbled upon words like "$\mathbb{Z}$-grading" and "standard grading", can anyone explain this?
In general you can take the grading of a ring to be any monoid $M$. This means that the ring $R$ can be written as a direct sum $R = \bigoplus_{x \in M} R_x$, where each $R_x$ is an abelian subgroup of $R$, and $R_x R_y \subseteq R_{x \cdot y}$ (where $\cdot$ is the multiplication in $M$). Such a ring is called an "$M$-graded ring".
In the first case Atiyah considers ring graded by $\mathbb{N} = \{0, 1, \dots\}$, but you can also find rings graded by $\mathbb{Z}$ as in your second example. Any ring graded by $\mathbb N$ is also graded by $\mathbb Z$, by the way, by letting $R_i = 0$ for $i < 0$.