Circle
Elementary
A circle is the round curve made of all the points at a given distance from a single point. The distance is called the radius and the point at the middle is called the center.
A circle is only the curve, not the region inside it.
High school
Consider a center $c$ in the plane and a radius $r > 0$. The circle with center $c$ and radius $r$ is the set of points in the plane at distance $r$ from $c$. It is a sphere in $\reals^2$ (the $1$-sphere $S^1$).
Examples
Non-examples
Things that may be mistaken for examples.
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The boundary of a square
High school
The boundary of a square is not a (geometric) circle, because its points are not all at the same distance from the center.
Notes
The region inside a circle is a disk. “Find the area of the circle” really means the area of the disk.