The term nostro account is a bit of an umbrella term and can have different meanings for different contexts. For instance, any regular demand deposit account might be called a nostro account by the customer since he retains a claim on the funds after storing it with the bank; the bank calls it a vostro account since it is only holding the funds for the customer. A more proper usage of the term applies to interbank relationships, where one bank maintains an account at a foreign bank denominated in the foreign currency. The depositing bank calls it nostro; the foreign bank refers to it as vostro.
Nostro and Vostro
Nostro and vostro are actually versions of the Latin words meaning "ours" and "yours" respectively. Modern retail banking has its roots in 13th and 14th century Italy, where both depositors and retail banks maintained ledgers of their account balances. The ledger kept by the depositing customer referred to it as a nostro ledger, while the bank maintained the corresponding vostro ledger.
Over time, however, customers stopped maintaining their own ledgers. The few banking relationships that retained the practice tended to involve multiple banks. For instance, one bank might not want to go through the hassle of entering the forex market whenever a customer needs currency exchanged, so it could maintain a balance in foreign currency with a foreign bank. This is where nostro and vostro accounts predominantly exist.
Differences with Standard Bank Accounts
Other than the title, there are two main differences between a nostro account and a standard demand deposit account. The first is that nostro accounts tend to be held by financial institutions, while standard accounts are held by a wide variety of economic actors. The second is that nostro accounts are denominated in foreign currencies; most regular bank accounts use domestic currency.