A bank run is the most common historical example of a systemic risk. The most recent example of this is in this current issues syllabus—The Northern Rock bank run.
Banks borrow short-term from depositors and lend long-term to borrowers. Should depositors lose confidence in the bank and demand their deposits back in enough numbers, the bank has a severe liquidity problem.
In the US, Federal Deposit Insurance has largely eliminated this systemic risk.