If you mean that all the vertices of the polygon touch the circle, it is called an inscribed polygon.
You might also be thinking of a cyclic quadrilateral:
see http://en.wikipedia.org/wiki/Cyclic_quadrilateral.
A circle with a polygon in it An inscribed polygon is any polygon that can fit within a specific curve or circle.
A regular polygon
A regular polygon
the circle is tangent to each side of the polygon And it's located within the polygon
Inscribed polygon, since it is inside the circle.
When a polygon is within a circle and the circle touches each one of its corners it is referred to as circumscribed.
An inscribed polygon
A square or an equilateral triangle for example when a circle is inscribed within it.
No, a circle is not a polygon
A circle is not a polygon.
A polygon whose vertices are on a circle and whose other points are inside the circle is called a "cyclic polygon." The circle is known as the circumcircle of the polygon, and all the vertices lie on its circumference. In addition to the vertices, the polygon may have additional points that are located within the circle, but those points do not change the cyclic nature of the polygon. Examples include triangles, quadrilaterals, and other polygons as long as their vertices are on the circle.
A circle can be a polygon. Sometimes a circle can be a polygon that has infinite number of sides.