Definition:Geometric Figure

A geometric figure is intuitively defined as a set of points and lines in space.

As defined by Euclid:

"A figure is that which is contained by any boundary or boundaries."

The boundary may or may not be included in a particular figure. If this is important (and in the study of topology it usually is), then whether it is included or not needs to be specified.

Plane Figure
A plane figure is a geometric figure embedded in $$\mathbb{R}^2$$.

Rectilineal Figure
As Euclid defined it:

" Rectilineal figures are those which are contained by straight lines."

It follows from this definition that a rectilineal figure is a plane figure.