A curve is a line which may or may not be straight.
A plane curve is a curve which can be embedded in the plane.
A skew curve is a curve which cannot be embedded in the plane.
A space curve is a curve which is embedded in $3$-space, and may or may not be a skew curve.
An arc is a specific part of a curve.
The term curve can often be seen for curved line, but is then specifically understood to mean a line which is specifically not straight.