Answer
parabola, directrix is perpendicular to the polar axis, $1$ unit(s) to the right of the pole, vertex $(\frac{1}{2},0)$
See graph.
Work Step by Step
1. Given $r=\frac{1}{1+cos\theta}$, we have $e=1$, thus it is a parabola. From the equation and with the $cos\theta$ term, we know $p=1$ and the directrix is perpendicular to the polar axis, $1$ unit(s) to the right of the pole, vertex $(\frac{1}{2},0)$
2. See graph.
