Gaussian covariance matrix basic concept We have a $$ \boldsymbol\mu= \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix} $$
$$ \boldsymbol Y= \begin{bmatrix} Y_1 \\ Y_2 \end{bmatrix} $$
with 
$$ 
\begin{bmatrix}
\Sigma_{11} & \Sigma_{12}\\
\Sigma_{21} & \Sigma_{22}
\end{bmatrix}
$$
Why they represent covariance with 4 separated matrix? 
and what does  $$ \Sigma_{11}  \Sigma_{12}  \Sigma_{21}  \Sigma_{22}  $$
seperately mean in Gaussian? 
 A: 
Why they represent covariance with 4 separated matrices?
  I emphasize this each notion as matrix. what happen if each notion become a matrix 

In this case the vectors ${\boldsymbol Y}$ and ${\boldsymbol \mu}$ are really block vectors. In the case of an $n$-dimensional ${\boldsymbol Y}$ vector we could expand it as follows:
$$\boldsymbol Y= \begin{bmatrix} \color{blue}{Y_1} \\ \color{red}{Y_2} \end{bmatrix}=\begin{bmatrix}\color{blue}{Y_{11}\\Y_{12}\\\vdots\\ Y_{1h}}\\\color{red}{Y_{21}\\Y_{22}\\\vdots\\ Y_{2k}}\end{bmatrix}\tag{$n \times 1$}$$
showing the partition of the $n$ coordinates into two groups of size $h$ and $k$, respectively, such that $n = h + k$. A parallel illustration would immediately follow for the $\boldsymbol \mu$ vector of population means.
The block matrix of covariances would hence follow as:
$$\begin{bmatrix}
\Sigma_{\color{blue}{11}} & \Sigma_{\color{blue}{1}\color{red}{2}}\\
\Sigma_{\color{red}{2}\color{blue}{1}} & \Sigma_{\color{red}{22}}
\end{bmatrix} \tag {$n \times n$}$$
where
$$\small\Sigma_{\color{blue}{11}}=\begin{bmatrix}
\sigma^2({\color{blue}{Y_{11}}}) & \text{cov}(\color{blue}{Y_{11},Y_{12}}) & \dots & \text{cov}(\color{blue}{Y_{11},Y_{1h}}) \\
\text{cov}(\color{blue}{Y_{12},Y_{11}}) & \sigma^2({\color{blue}{Y_{12}}}) & \dots & \text{cov}(\color{blue}{Y_{12},Y_{1h}}) \\
\vdots & \vdots & & \vdots \\
\text{cov}(\color{blue}{Y_{1h},Y_{11}})  &  \text{cov}(\color{blue}{Y_{1h},Y_{12}}) &\dots& \sigma^2({\color{blue}{Y_{1h}}})
\end{bmatrix} \tag{$h \times h$}$$
with 
$$\small \Sigma_{\color{blue}{1}\color{red}{2}}=
\begin{bmatrix}
\text{cov}({\color{blue}{Y_{11}}},\color{red}{Y_{21}}) & \text{cov}(\color{blue}{Y_{11}},\color{red}{Y_{22}}) & \dots & \text{cov}(\color{blue}{Y_{11}},\color{red}{Y_{2k}}) \\
\text{cov}({\color{blue}{Y_{12}}},\color{red}{Y_{21}}) & \text{cov}(\color{blue}{Y_{12}},\color{red}{Y_{22}}) & \dots & \text{cov}(\color{blue}{Y_{12}},\color{red}{Y_{2k}}) \\
\vdots & \vdots & & \vdots \\
\text{cov}({\color{blue}{Y_{1h}}},\color{red}{Y_{21}}) & \text{cov}(\color{blue}{Y_{1h}},\color{red}{Y_{22}}) & \dots & \text{cov}(\color{blue}{Y_{1h}},\color{red}{Y_{2k}})
\end{bmatrix}\tag{$h \times k$}
$$
its transpose...
$$\small \Sigma_{\color{red}{2}\color{blue}{1}}=
\begin{bmatrix}
\text{cov}({\color{red}{Y_{21}}},\color{blue}{Y_{11}}) & \text{cov}(\color{red}{Y_{21}},\color{blue}{Y_{12}}) & \dots & \text{cov}(\color{red}{Y_{21}},\color{blue}{Y_{1h}}) \\\text{cov}({\color{red}{Y_{22}}},\color{blue}{Y_{11}}) & \text{cov}(\color{red}{Y_{22}},\color{blue}{Y_{12}}) & \dots & \text{cov}(\color{red}{Y_{22}},\color{blue}{Y_{1h}}) \\
\vdots & \vdots & & \vdots \\
\text{cov}({\color{red}{Y_{2k}}},\color{blue}{Y_{11}}) & \text{cov}(\color{red}{Y_{2k}},\color{blue}{Y_{12}}) & \dots & \text{cov}(\color{red}{Y_{2k}},\color{blue}{Y_{1h}})
\end{bmatrix}\tag{$k \times h$}
$$
and
$$\small \Sigma_{\color{red}{22}}=\begin{bmatrix}
\sigma^2({\color{red}{Y_{21}}}) & \text{cov}(\color{red}{Y_{21},Y_{22}}) & \dots & \text{cov}(\color{red}{Y_{21},Y_{2k}}) \\
\text{cov}(\color{red}{Y_{22},Y_{21}}) & \sigma^2({\color{red}{Y_{22}}}) & \dots & \text{cov}(\color{red}{Y_{22},Y_{2k}}) \\
\vdots & \vdots & & \vdots \\
\text{cov}(\color{red}{Y_{2k},Y_{21}})  &  \text{cov}(\color{red}{Y_{2k},Y_{22}}) &\dots& \sigma^2({\color{red}{Y_{2k}}})
\end{bmatrix} \tag{$k \times k$}$$
These partitions come into play in proving that the marginal distributions of a multivariate Gaussian are also Gaussian, as well as in the actual derivation of marginal and conditional pdf's.
