\mid causes tex to switch into math mode, and it's still in math mode when it gets to the next \item, hence the unexpected error message.
instead of what you have, make the line with \mid an unnumbered math display:
\item \textbf{Give a regular expression for the language accepted by the automaton~$A$.}
\[ (0 \mid 1)*01)0 \mid 1)* \]
\item \textbf{Explain what makes the automaton~$A$ non-deterministic.}
The reason automaton~$A$ is non-deterministic is because in the state $q_0$ reading a $0$
can lead to either $q_0$ or $q_1$.
using double backslashes, as already commented, is not a good practice. see
When to use \par and when \\
for more information.
edit: based on a comment about what is math and what isn't, i've adjusted the original
code; i've also applied my "editing pencil" to the position of "either" for clarity.
(apologies for my presumption; see my user profile to understand why i'm fussy this way.)
\midmust appear in math mode and it produces errors in normal text modes. – egreg Feb 20 '15 at 14:54\\in the text, ever! – daleif Feb 20 '15 at 14:55\textbarmacro to typeset a vertical bar in textmode. I'd just put the regex in math mode though:\[(0 \mid 1)*01)0 \mid 1)*\](removing the\\) – TonioElGringo Feb 20 '15 at 15:16