What would be an example of a proof system being sound but not complete? I know "p ^ not p" would be complete but not sound, but what would be a simple example of a system being sound but not complete? 
 A: The most simple example of a proof system that is sound but not complete would be the one that has no inference rules at all!
It proves nothing except whichever non-logical axioms your theory has, so in particular it doesn't prove anything that risks being false -- so it is sound. But it is very much not complete.

A more interesting example is intuitionistic logic. Everything it proves is also a theorem of classical propositional logic (so it is sound if classical logic is), but it is not complete with respect to the classical semantics of the propositional calculus. For example, intuitionistic logic does not prove $p\lor \neg p$, even though this formula evaluates to true under every truth assignment.
Many (but not all) proof systems for the propositional calculus are deliberately structured such that one gets intuitionistic logic by removing or modifying just one rule. If you have either of


*

*Peirce's law: $((p\to q)\to p)\to p$

*Excluded middle: $p\lor\neg p$

*Double-negation elimination $\neg\neg p \to p$

*Proof by contradiction $(\neg p\to q\land \neg q)\to p$


among your logical axioms, then it is likely that removing that axiom will give you intuitionistic logic.
(One can define other kinds of semantics that intuitionistic logic is complete with respect to, however).
