Universality of the Toffoli gate
Toffoli is universal for classical computation (as shown by @Victor). However, Toffoli is NOT universal for quantum computation (unless we have something crazy like $P = BQP$).
To be universal for quantum computation (under the usual definition), the group generated by your gates has to be dense in the unitaries. In other words, given an arbitrary $\epsilon$ and target unitary $U$ there is some way to apply a finite number of your quantum gates to get a unitary $U'$ such that $||U - U'|| &lt; \epsilon$.
Toffoli by itself is clearly not universal under this definition since it always takes basis states to  basis states, and thus, for example, it cannot implement something that takes $|0\rangle \rightarrow \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$. In other words, it cannot create superposition.