What's the difference (notation and meaning) for the following variance and estimators 
*

*variance of estimator $\quad\cdots\quad$ I believe this is $\quad \text{Var}(\bar{y})$

*estimated variance $\quad\cdots\quad$ I believe this is $\quad \widehat{\text{Var}}(\bar{y})$

*estimated variance of estimator...no clue
 A: In the context of survey sampling, mostly, the population mean is denoted by $\bar{Y}$ and its estimator  by $\bar{y}$, a random variable. Note that, the symbol $\bar{y}$ is also used to denote the estimate, a numerical quantity obtained from the sample. 
$\text{Var}(\bar{y})$ is used  to refer to the variance of the estimator. It is a parameter of the sampling distribution of the estimator $\bar{y}$, which is defined in terms of the population parameter. It is unknown. An estimate of this is denoted by $\widehat{\text{Var}}(\bar{y})$,  a numerical quantity computed from the sample and is referred to as estimated variance of the estimator or simply estimated variance.
In SRSWOR, for example, 
\begin{equation*}
\text{Var}(\bar{y})=\left(\dfrac{N-n}{N}\right)\dfrac{S^2}{n}.
\end{equation*}
 It can be shown that the sample variance
\begin{equation*}
s^2 = \dfrac{1}{n-1}\sum_{i=1}^{n}(y_{i}-\bar{y})^2
\end{equation*}
is an unbiased estimator for the population mean square $S^2$. Thus, the estimated variance of the sample mean is obtained by replacing $S^2$ by $s^2$ in $\text{Var}(\bar{y})$. That is,
\begin{equation*}
\widehat{\text{Var}}(\bar{y})= \left(\dfrac{N-n}{N}\right)\dfrac{s^2}{n}.
\end{equation*}
A: Example:
Let $Y_i,  i = 1,...,n$ are iid following $N(\mu,\sigma^2)$.
$\bar Y = \frac{\sum_{i=1}^n Y_i}n$ be the estimator of $\mu$.
Then variance of estimator $\bar Y$ is $\sigma^2/n$
The estimated variance is $\hat \sigma^2 = \frac{\sum_{i=1}^n (Y_i-\bar Y)^2}{n-1}$
The estimated variance of estimator $\bar Y$ is $\hat \sigma^2/n$
Difference situation may have different explanation. Need to follow the text of the paper/book.
