Differences between best response, dominant strategy and Nash equilibrium
Consider the following game between P1 (row player) and P2 (column player):
\begin{array}{|c|c|c|}\hline
&amp; L &amp; R \\\hline
T&amp; 1,1 &amp; 2,0 \\\hline
B&amp; 0,0 &amp; 1,1 \\\hline
\end{array}

$T$ is P1's dominant strategy
$T$ is P1's best response to both of P2's strategies $L$ and $R$
$L$ is P2's best response to P1's strategy $T$
$R$ is P2's best response to P1's strategy $B$
$(T,L)$ is the only Nash equilibrium

Generalizing from the above observations:

A strategy is dominant if and only if it is a best response to each of the other player's strategies, e.g. P1's $T$.
A strategy that is a best response to some but not all of the other player's strategies cannot be a dominant strategy, e.g. P2's $L$ and $R$.
A Nash equilibrium is a pair of mutually best responding strategies, e.g. $(T,L)$.