I have the following FSA, and the regular language that I found for it:

Is this language correct? It doesn't match the solution in the book, but my teacher says there can be multiple equally correct languages. The book's solution looks like this: $(0\cup1)1^{*}00^{*}(11^{*}00^{*})^{*}$. Are these two languages equal?
Yes, they are. You can simplify both to $$\Sigma^+\mathtt{0}$$ for $\Sigma = \{\mathtt{0},\mathtt{1}\}$, that is, $$(\mathtt{0}\cup\mathtt{1})(\mathtt{0}\cup\mathtt{1})^*\mathtt{0}$$
The crucial part is to observe that, after you cleared the initial state (i.e. read the first symbol), whatever you do, and wherever you are, after reading symbol $\mathtt{0}$ you get to the accepting state.
The difference between your solution and the solution from your book is that
These are equivalent because both are in a loop, i.e. $B^*(ABB^*)^* = (AB \cup B)^*$.
I hope this helps $\ddot\smile$