An overdraft is when you draw more money from your account than you actually have; i.e., when you go into a debit balance.
Overdrafts can be divided into two catergories, authorised and unauthorised. An authorised overdraft normally works like a small unsecured loan allowing you to go overdrawn on your current account up to a certain limit.
An authorised overdraft differs from a loan in that is accessed via the current account, rather than a specific loan acccount, is generally only for a small amount (around £5,000) and the has no requirements for repayment and what it is used for.
An unauthorised overdraft is when you go into a debit balance on your account. This is most commonly caused by a transaction going through on a debit card when the account limit is not completely up to date. With debit cards such as Visa Delta or the Switch card the account is not immediately debited when a transaction or withdrawl takes places so it is possible for the balance to be displayed incorrectly even on an ATM.
If you go overdrawn without authorisation then the bank is likely to charge you heavily. Generally they will charge you a set fee for going overdrawn and then a high rate of interest on the outstanding debit.
Many banks will allow you to set up a zero interest no fee overdraft at a low level. Other will allow a low interest overdraft. Many people take this up in case of need whilst others like to avoid the temptation of an overdraft.
An alternative to the heavy charges of an high street bank is to use a private bank such as SG Hambros. These banks will not charge you for going overdrawn unauthorised and will be more flexible about cashing cheques and allowing credit card transactions when you are already at your limit (be it 0 or an authorised overdraft limit). However private banks require their customers to be wealthy people, or "High Net Worth Individuals". This means they need to have minimum liquid assets of £250,000. In addition they pay for the service of the bank overall rather than paying for individual services such as overdrafts.