Why is a circle 1-dimensional? In the textbook I am reading, it says a dimension is the number of independent parameters needed to specify a point. In order to make a circle, you need two points to specify the $x$ and $y$ position of a circle, but apparently a circle can be described with only the $x$-coordinate? How is this possible without the $y$-coordinate also?

You need not getting too sophisticated. 
One dimension means "forward or backward" - these are your options if you sit on a line.
A circle is just a bent line in a way that you reach the same spot after some distance.
So a circle is finite but unlimited.
If you think about the "bending" it is quite interesting that this bending takes place in the second dimension.
Take this one level higher:
in two dimensions your options are "forward, backward, left and right"
so on a plane you have these choices.
Now bend your plane in a special way... the corresponding thing to the circle is a sphere.
Next level... out of our imagination ;-)
