Answer
Parabola
The directrix is perpendicular to the polar axis at a distance of one unit to the right of the pole.
Work Step by Step
We are given the equation in polar coordinates:
$r=\dfrac{1}{1+\cos\theta}$
The equation is in the form:
$r=\dfrac{ep}{1+e\cos\theta}$
Identify $e$ from the denominator, then $p$ from the numerator:
$e=1$
$ep=1\Rightarrow (1)p=1\Rightarrow p=1$
Because $e=1$, the conic is a parabola. The directrix is perpendicular to the polar axis at a distance of one unit to the right of the pole.