What figure is described below?

the points in space equidistant from the endpoints of pq.

The question is slightly weird.

You have a piece of line, and its ends are 'P' and 'Q'.

All of the points on the paper that are the same distance
from 'P' and 'Q' are the points that form the line that bisects
segment-PQ and is perpendicular to it.
Other words:  The line that is perpendicular to PQ at its mid-point.
Other words:  The perpendicular bisector of segment PQ.

All of the points in space that are the same distance from
'P' and 'Q' are the points that form the plane perpendicular
to segment PQ at its mid-point.