I need to clear up my confusion on the definition of a smooth manifold. So we say that $M$ is a smooth manifold (of dimension $n$), if $M$ is Hausdorff and if every $x \in M$ is contained in a neighborhood $U$ that's homeomorphic to an n-ball (the pair $\phi, U$ is called a chart), and if two such charts $\phi_1, U_1$, and $\phi_2, U_2$ overlap, then
$$\phi_2 \circ \phi_1^{-1} : \phi_1(U_1 \cap U_2) \to \phi_2(U_1 \cap U_2)$$
is a smooth map.
But I remember my professor proving that a certain space was a smooth manifold by merely finding an atlas (an open covering of the space by charts) such that the above holds. But according to the definition I wrote, this would be insufficient. Can anyone clear up my confusion?
I think your definition of a manifold is missing something. A topological manifold is a Hausdorff space $M$ such that every $x\in M$ has a neighbourhood $U$ that is homeomorphic to some open subset of $\mathbb R^n$. (Usually we also assume paracompactness.) The pair $(\phi,U)$ where $\phi$ is the homeomorphism I mentioned is called a chart.
Now the main point: to specify a smooth structure on $M$ we have to specify which charts we consider to be smooth. So, a smooth manifold is a topological manifold together with a set of smooth charts, which is a subset of all charts. And it is these smooth charts that we require to be smoothly compatible, i.e. for any two smooth charts $(\phi,U)$, $(\psi,V)$, we require that $$\psi\circ\phi^{-1}:\phi(U\cap V)\to\psi(U\cap V)$$ is smooth. A collection of such smooth charts that covers $M$ is called a smooth atlas.
(The situation is a bit similar to the one in the definition of a topological space: a topological space is a space in which we specify which subsets are open. A smooth manifold is a manifold in which we specify which charts are smooth.)