In class field theory, we have the well-known local invariant map $\mathrm{inv}_v: \mathrm{Br}(k_v) \rightarrow \mathbb{Q}/\mathbb{Z}$, where $k$ is a field and $v$ is a place of $k$. Similarly, we know that closed points on a smooth curve correspond to places of its function field, so it is not surprising that we have an analogous map. First, we fix our setting.
Let $k$ be a number field and $X$ a smooth projective curve of genus $\geq 1$. Let $K$ denote the function field of $X$. We have an exact sequence (see for example, Milne's EC), $$0 \rightarrow \mathrm{Br}(X) \rightarrow \mathrm{Br}(K) \xrightarrow{\mathrm{inv}}\bigoplus _{x \in X^{(1)}} H^1(k(x),\mathbb{Q}/\mathbb{Z}),$$ where $X^{(1)}$ is the set of closed points of $X$ and $k(x)$ is the residue field of $x$, i.e., a finite field extension of $k$.
My reference is the paper Brauer groups of curves and reciprocity laws in Brauer groups of their function fields by V. I. Yanchevskii. He wrote the following about the map inv:
The homomorphism inv is induced by the local ones $\mathrm{inv}_x: \mathrm{Br}(K) \rightarrow H^1(k(x),\mathbb{Q}/\mathbb{Z})$. If $a \in \mathrm{Br}(K)$, then the character $\mathrm{inv}_x(a)$ is called the local invariant of $a$ at $x$ and its order equals to the ramification index at $x$ of algebras from $a$.
What does "...its order equals to the ramification index at $x$ of algebras from $a$" mean here? I know that $a$ is a class of central simple algebras over $K$ but I don't know what does ramification index at $x$ mean here. Maybe a concrete example, or a reference to an example, will help.