How and in what context are polynomials considered equal? There's two notions of equivalent polynomials floating around, one saying that $f = g$ iff they're equivalent as maps, and the other saying $f = g$ iff they're equal on each coefficient when written in standard form.  
I'm interested in polynomials over a finite field, irreducible polynomials and factoring so what type of equivalence should I use?  For instance if we take map equivalence, then there are only a finite number of polynomials.  And that makes a huge difference!
Please explain when it's okay to use what.

In abstract algebra polynomials over a ring $R$ are defined as formal sums
$$
\sum_{k=0}^N a_k X^k
$$
where $X$ is a formal variable and all $a_k\in R$. To make this precise, we can also model polynomials as sequences $(a_0, a_1, \dots)$ where all but finitely many $a_i$ are zero. Addition and multiplications is then given by
$$
(a_0, a_1, \dots) + (b_0, b_1, \dots) := (a_0+a_1, b_0+b_1, \dots)
$$
and
$$(a_0, a_1, \dots) \cdot (b_0, b_1, \dots) := (c_0, c_1, \dots),\quad\text{where}\ c_k = \sum_{i=0}^k a_i b_{k-i}.$$
This corresponds to the usual coefficient-wise addition of polynomials and the Cauchy product formula
$$
\left(\sum_{k=0}^\infty a_k X^k \right) \cdot \left(\sum_{k=0}^\infty b_k X^k \right) = \sum_{k=0}^\infty c_k X^k,\quad\mathrm{where}\ c_k=\sum_{i=0}^k a_i b_{k-i}.
$$
Letting $X=(0,1,0,\dots)$ we obtain the usual representation of polynomials. So two polynomials are equal if the underlying tuples are equal which is the case if and only if all coefficients are equal.
Note that we can easily drop the "all but finitely many $a_i=0$" requirement and still get a ring structure with the above operations. This is the ring $R[[X]]$ of formal power series, while the ring of polynomials is usually denoted $R[X]$.
