How can there be an electronic bank run? I read that at 2008 there was an electronic bank run.

From what I undestand bank runs occur due to the fractional reserve banking system. The actual real money you can hold on your hand are a fraction of the total money in banks. So if everyone goes to ATM and gets the money in their hand, it means that the bank runs out of the "real money" and people can't withdraw these "real money".

So if the same number of people, instead of withdrawing from the ATMs, do electronic transactions that eg get their money from a bank to another bank (or any other institution) how can there be a bank run? It just doesn't make sense to me.
What you describe is the "traditional" bank-run.  

The 2008 incident you refer to, was called an "electronic bank run", because the problem was not caused by people trying to get tangible paper money in their hands in excess of the actual paper money available to the banks.  

The problem was that through electronic banking and electronic wire-transfer orders "funds were redirected out of the U.S. money markets" as one account of the incident puts it.  

So why is this a problem? 

Because under "fractional banking", electronic money functions exactly as paper money. You work, and your employer pays your salary through a wire transfer. You go buy something and you pay with your debit card, where you transfer electronic money from your account to the shop's account.  

In other words, electronic money, once created, is as much actual purchasing power as paper money is. It is wealth.

But this means that this "electronic bank run" was impoverishing the US banking system. But since the transfers were electronic, by necessity they went out of the US economy also -to some bank accounts in other countries.  It was wealth going out of the US economy, not just paper money going out of the banks but still in the economy -and at a speed and volume unimaginable in the old days, due to the wonders of the internet and digital technology.  

So essentially, it is a totally different kind of problem (and more serious) than the "traditional" bank run, so it should perhaps have been characterized by using another term.