Let $F$ be a field. Consider the set $$R = \left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} : a, b, c \in F \right\}$$ Think of a non-trivial two-sided ideal of $R$ and describe in a concrete way the quotient ring $R/I$. (Your description should be explicit enough to make the fact that $R/I$ is a commutative ring completely transparent.)
I can think of some ideals of $R$, however I don't understand the meaning of a quotient ring when it comes to matrices nor how to describe them.
The meaning of a ring is no different when the elements of the ring are matrices (or vectors, or elephants, or ...).
Given a ring $R$ and an ideal $I \trianglelefteq R$ the quotient $R/I$ is given by $$R/I = \{ x + I : x \in R \}$$ with addition and multiplication defined by $(x+I)+(y+I) = (x+y)+I$ and $(x+I) \cdot (y+I) = (x \cdot y) + I$.
Here it's exactly the same. If $I \trianglelefteq R$ is an ideal then $$R/I = \left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} + I : a,b,c \in F \right \}$$ To make it really explicit, notice that $$\begin{bmatrix} a & b \\ 0 & c \end{bmatrix} + I = \left\{ \begin{bmatrix} a + x & b + y \\ 0 & c + z \end{bmatrix} : \begin{bmatrix} x & y \\ 0 & z \end{bmatrix} \in I \right\}$$