For $A = \mathbb{F}_q[t]$, what is the meaning of $f\in A[x]$? 
Suppose $A = \mathbb{F}_q[t]$. What then is the meaning of $\ f\in A[x]$?.

My thoughts
To construct $\mathbb{F}_q = \mathbb{F}_{p^m}$ we take an irreducible polynomial $P$ of degree $m$ with coefficients in $\frac{\mathbb{Z}}{p\mathbb{Z}}$ and then define $$\mathbb{F}_{p^m} :=\frac{\frac{\mathbb{Z}}{p\mathbb{Z}}}{\langle P(x)\rangle}.$$
So for some $a \in \mathbb{F}_q[t]$, we have a polynomial whose coefficients are itself polynomials (both in $t$)? But what now for $f\in A[x]$? Could you provide an example of what the elements of $A[x]$ look like, say of degree 2 or 3?
 A: First of all, you should write $\mathbb{F}_{p^m} :=\dfrac{\frac{\mathbb{Z}}{p\mathbb{Z}}[X]}{\langle P(x)\rangle},$ you just forget "bracket $X$".
An element $a \in \mathbb{F}_q[t] = A$ is a polynomial in the variable $t$ whose coefficients are elements in $\Bbb F_q$, i.e. equivalence class of polynomials in $\Bbb F_p[X]$.
You should think about the following cases to see what happens :
1) if $m=1$, then $\Bbb F_q = \Bbb F_p$, so $A$ is just $\Bbb F_p[t]$.
2) Consider $\Bbb C \cong \Bbb R[X] / (X^2+1)$. Then an element $a \in \Bbb C[t]$ is a polynomial in the variable $t$ whose coefficients are complex numbers, i.e. equivalence classes of polynomials in $\Bbb R[X]$. Typically,
$$a = t^2  -  i t + 2+i = t^2 - [X]t + [2+X],$$
is an element of $\Bbb C[t]$ which has degree $2$.
When $A = \Bbb F_q[t]$, the elements of $A[Y]$ are polynomials in the variable $Y$ whose coefficients are elements of $A$. For instance,
$$w = a_2 Y^2 + a_1 Y + a_0$$
is element of $A[Y]$ with degree $2$, where $a_0, a_1, a_2 \in A = \Bbb F_q[t]$, e.g. $a_0 = a_1 = a_2 = t + 1$ where $1$ denotes the element $[1]_{P} \in \Bbb F_{p^m}$.
