Adjacent vertices: side
Non-adjacent: diagonal
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A line segment in a polygon connecting any two nonconsecutive vertices is called a diagonal.
If I understand what you mean, then another way to say is "No two of its vertices can form a line segment that falls outside of the polygon". In that case, it is called a convex polygon.
Yes, such a polygon is called a triangle.
It is a dodecagon.
A octagon