What is money really? Say I have €1000 (or $1000) on a bank account. What is that money really?

Is it a value stored on some computer at the bank where I have the bank account, a value stored at the central bank of the currency (in the case of the euro the European Central Bank) or something else?

And if it is just a value stored on a computer somewhere, why do we give it so much worth?
Money is, in essence, debt. More broadly, it's a system of clearing and credit. That entry in the computer means that the bank owes you \$1000, which is worth whatever others are willing to exchange for that sum— one way of thinking about it is that if you have \$1000, you have a general claim on the rest of society for \$1000 worth of whatever society produces. 

The fact is that most money is in the form of loans (debt), not currency. This is well-explained in a recent Bank of England paper.