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?

That is a bad definition of dimension. By this definition, the (filled-in) unit square in the real coordinate plane is one-dimensional, since you only need one number to describe a point $(x,y)$. An easy way to do this is to write out the decimal expansions of $x$ and $y$:
$$x=0.a_1a_2a_3a_4a_5\dots$$
$$y=0.b_1b_2b_3b_4b_5\dots$$
Now the point $(x,y)$ can be described with just one number:
$$0.a_1b_1a_2b_2a_3b_3\dots$$

But suppose we overlook this shortcoming of your definition, and we apply it to a circle. Then we can use the polar coordinate map
$$(x,y) \rightarrow (r,\theta)$$
This lets us write every point $(x,y)$ in terms of just one variable: $\theta$ (since $r$ is constant). 
