First Isomorphism Theorem examples
First Isomorphism Theorem (FIT) applies in different contexts: Groups, Rings, Vector Spaces, Lie Algebras and other structures.
As follows, examples for the first three.
Three Group Isomorphisms
Let us consider $GL_2(\Bbb F_3)$: the group of $2\times 2$ invertible matrices with coefficients in $\Bbb F_3$ which is the field with three elements. This is a group wrt the matrix multiplication. Consider now the linear map $\det:GL_2(\Bbb F_3)\to\Bbb F_3^{\times}$. It's clearly surjective, and its kernel are all the matrices with determinant equal to $1$, which form a normal subgroup denoted by $SL_2(\Bbb F_3)$.
Thus from FIT it follows that
$$
\color{#c01}{\frac{GL_2(\Bbb F_3)}{SL_2(\Bbb F_3)}\simeq\Bbb F_3^{\times}}\;.
$$
Another interesting example which descend from FIT is the following (but proving it is not immediate): let $\Bbb F$ be a field, and $t$ a trascendental element over $\Bbb F$. It can be proven that
$$
\color{#b49}{\operatorname{Aut}(\Bbb F(t)|\Bbb F)\simeq GL_2(\Bbb F)/D_2(\Bbb F)}
$$
where $D_2(\Bbb F):=\{a\Bbb I_2\;:\;a\in\Bbb F\}$ are the $2\times2$ scalar matrices.
A last interesting example with groups: consider the additive group $H:=(\Bbb Z/m\Bbb Z,+)$. Then
$$
\color{#d78}{\operatorname{Aut}(H)\simeq U(\Bbb Z/m\Bbb Z)}
$$
where $U(\Bbb Z/m\Bbb Z)$ is the group of unit (i.e. the invertible elements) of the ring $\Bbb Z/m\Bbb Z$. Try to figure out the morphism between the two groups and check it is bijective.
A Ring Isomorphism
Consider a field $\Bbb K$, $\operatorname{char}(\Bbb K)=0$ and $\overline {\Bbb K}=\Bbb K$ (i.e. algebraically closed).
Consider the ring morphism given by $\varphi:\Bbb K[X,Y,Z]\to\Bbb K[X,Z]$ given by $X\mapsto X$, $Z\mapsto Z$ and $Y\mapsto XZ$.
Then $\varphi$ is clearly surjective, and the kernel is the ideal $(ZX-Y)$. Thus FIT gives
$$
\color{#c33}{\frac{\Bbb K[X,Y,Z]}{(ZX-Y)}\simeq\Bbb K[X,Z]}\;.
$$
A Vector Space Isomorphism
Consider the map $\phi:\Bbb R^4\to\Bbb R^2$ given by $\begin{bmatrix}x\\y\\z\\w \end{bmatrix}\mapsto\begin{bmatrix}x-z\\y+w \end{bmatrix}$.
$\phi$ is an $\Bbb R$-linear map, clearly surjective, in which the kernel is given by
$
W:=\langle\begin{bmatrix}1\\0\\1\\0\end{bmatrix},
\begin{bmatrix}0\\1\\0\\-1 \end{bmatrix}\rangle
$.
Hence, FIT gives
$$
\color{#c03}{\Bbb R^4/W\simeq\Bbb R^2}\;\;.
$$