  Several upper and lower bounds for the numerical radius of $2 \times 2$
operator matrices are developed which refine and generalize the earlier related
bounds. In particular, we show that if $B,C$ are bounded linear operators on a
complex Hilbert space, then \begin{eqnarray*}
  && \frac{1}{2}\max \left \{ \|B\|, \|C\| \right \}+\frac{1}{4} \left |
\|B+C^*\|-\|B-C^*\| \right |
  &&\leq w \left(\left[\begin{array}{cc}
  0 & B
  C& 0
  \end{array}\right]\right)\\ &&\leq \frac{1}{2} \max \left\{\|B\|,\|C\|\right
\}+\frac{1}{2}\max \left
\{r^{\frac{1}{2}}(|B||C^*|),r^{\frac{1}{2}}(|B^*||C|)\right\}, \end{eqnarray*}
where $w(.)$, $r(.)$ and $\|.\|$ are the numerical radius, spectral radius and
operator norm of a bounded linear operator, respectively. We also obtain
equality conditions for the numerical radius of the operator matrix
$\left[\begin{array}{cc}
  0 & B
  C& 0 \end{array}\right]$. As application of results obtained, we show that if
$B,C$ are self-adjoint operators then, $\max \Big \{\|B+C\|^2 , \|B-C\|^2
\Big\}\leq \left \|B^2+C^2 \right \|+2w(|B||C|). $
