The correct answer is $\boxed{\text{C}}$.
To find the eigenvalues of a matrix, we can use the following formula:
$$\lambda = \det(P – \lambda I)$$
where $I$ is the identity matrix.
In this case, we have:
$$\begin{align}
\det(P – \lambda I) &= \det \left[ {\begin{array}{{20}{c}} 1&1&0 \ 0&1&1 \ 0&0&1 – \lambda \end{array}} \right] \
&= 1 – \lambda + \lambda^2 \
&= (\lambda – 1)^2
\end{align*}$$
Therefore, the eigenvalues of $P$ are $\lambda = 1$ and $\lambda = 1$. Since there are two distinct eigenvalues, the answer is $\boxed{\text{C}}$.
The other options are incorrect because they do not correspond to the number of distinct eigenvalues of $P$. Option $\text{A}$ is incorrect because there are two distinct eigenvalues, not three. Option $\text{B}$ is incorrect because there are two distinct eigenvalues, not zero. Option $\text{D}$ is incorrect because there are two distinct eigenvalues, not one.