Sorry for my poor English ,I don't know the precise mean of variations in the below picture .I think it like coordinate translation ,but I'm not sure .
Besides, if $\delta g_{ij}=v_{ij}$,how to compute $\delta g^{ij}$? I guess it should be using the $g_{ij}g^{ij}=1$, but I don't know what $\delta 1$ equal to .
Thanks very much for detail answer or hint .
Below picture is from the 199th page of this paper.
Using $g^{ij} g_{jk} = \delta_{ik}$, differentiate both sides to find $$g^{ij} \delta g_{jk} = -\delta g^{ij} g_{jk} \Rightarrow \delta g^{ij} = - g^{ij} v_{jk} g^{kl}. $$
In general, this $\delta$ is a common notation used in the calculus of variations. If $g_t$ is a family of (metrics, functions, embeddings...), then $$\delta g := \frac{\partial}{\partial t}\bigg|_{t=0} g_t.$$