What does the Circle really mean? 
Which of the following figure is really the circle? If a point is on the circle it means that point should be on the circumference is it? (point $Z$ on figure 1). 
Point $P$ on figure $1$ is inside the circle not on the circle. 
Which means the circle is like a ring with no inside. 
In figure $2$, the circle is cut from a paper and hence has an inside. 
There are two points $Q$ and $X$. 
$X$ is on the circumference so we can say it is on the circle. 
Now what a about the point $Q$ can we say it is on the circle? 
I am confused with the definition of circles.

Usually a circle is defined to be the set of all points $(x,y)$ in the plane that are a fixed distance from some point $p=(p_1,p_2)$ in the plane.  For instance $9=(x-p_1)^2+(y-p_2)^2$ describes the circle centered at $p$ with radius 3 (by the Pythagorean Theorem).  
The solid region is usually referred to as a disc, the set of all points in the plane that have distance from $p$ less than or equal to a given number.
