How many dimensions does a circle have?

A circle is a one-dimensional object, although one can embed it into a two-dimensional object. More precisely, it is a one-dimensional manifold. Manifolds admit an abstract description which is independent of a choice of embedding: for example, if you believe string theorists, there is a $10$- or $11$- or $26$-dimensional manifold that describes spacetime and a few extra dimensions, and we can study this manifold without embedding it into some larger $\mathbb{R}^n$.

Incidentally, you might guess that $n$-dimensional things ought to be embeddable into $\mathbb{R}^{n+1}$ ($n+1$-dimensional space). Actually, this is false: there are intrinsically $n$-dimensional things, such as the Klein bottle (which is $2$-dimensional) which can't be embedded into $\mathbb{R}^{n+1}$. The Klein bottle does admit an embedding into $\mathbb{R}^4$. More generally, the Whitney embedding theorem tells you that smooth $n$-dimensional manifolds can be embedded into $\mathbb{R}^{2n}$ for $n > 0$, and for topological manifolds see this MO question.

There are also other notions of dimension more general than that for manifolds: see the Wikipedia article.