I was reading a note about modular arithmetic, it says that if $x+y=0$,, then the additive inverse of $x$ is $y$, -1 mod 3=2, since 1+2=0. I'm confused, what does 1+2=0 related to -1 mod 3?
Another question is that you know we can check the table to get the additive inverse
for example, 1 mod 3=2 since 1+2=0 mod 3 where we can locate from the table, but how can we do the same thing on -1 mod 2=0, do we just get rid of "-" ?
Two numbers are equivalent modulo 3 if you can write one of them as a multiple of 3 plus the other. That is: $a\equiv b\mod 3$ means $a=3k+b$.
In the examples you gave we have: $(1+2)\equiv0\mod3$ because $1+2=3=3(1)+0$. So you can say that $2$ and $1$ are the additive inverse of each other in this structure.
A way to denote that is writing $2=-1$ but this one may lead to confusion. So another way to say the same thing is $-1\equiv2\mod3$ which means that $-1=-3+2=3(-1)+2$. In this case we can also verify that $-1$ and $-2$ are the additive inverses of each other.