  We construct rational elliptic surfaces of index two by explicitly
constructing their associated Halphen pencils in the projective plane
$\mathbb{P}^2$. For each of the types of singular fibers that occur we
construct at least one example having that type of fiber and in fact, for some,
we construct all possible examples. We establish a precise dictionary between
the fibers in a rational elliptic surface and the corresponding plane curves
and, in particular, we study the singularities of the curves appearing in a
Halphen pencil.
