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?

Alex Jordan gives the official answer if you officially define the circle as a differential manifold.  If you officially define it as a topological space then it is one dimensional because you can cut it into separate parts by removing single points.  Points are 0 dimensional, and connected spaces that can be disconnected by removing points are 1 dimensional.  
Actually this is local.  Two cones joined at the vertex can be disconnected by removing one point -- the vertex.  You could say a circle is one dimensional at every point, while those two cones have one dimension at the vertex but two dimensions elsewhere.
Or you could limit the definition to spaces that are locally isomorphic to $\mathbb{R}^n$.  That use of $n$ does not make the idea trivial.  Rather it shows that the topological criterion of connectedness agrees with the linear algebra criterion of a basis in defining dimension of the $\mathbb{R}^n$.
