In simple terms, an overdraft is the debt you go into if you spend more than you have in your current account. The moment your current account goes into the red, you step into your overdraft - unless of course your bank account does not allow any overdraft, in which case your payment will simply fail. Some accounts allow a small overdraft buffer, for example £10 or £20, before overdraft fees are charged.
To make things a bit more complicated, there are two types of bank overdraft. If your current account provider has previously agreed that you can go into debt, i.e. an overdraft, you will go into an arranged overdraft. Going over an arranged overdraft, or going into debt where there is no arranged overdraft, on the other hand, is called an unarranged overdraft.
Providers usually charge different fees for arranged and unarranged overdrafts; arranged overdrafts generally result in smaller fees, or even none at all. That's why you should always apply for an overdraft if you can, as unarranged overdraft charges can accumulate quite quickly.
For instance, if your provider charges you a fee of £5 per day for an unarranged overdraft and it takes you a week to pay off the debt, you would then owe your provider £35. Sometimes, this would happen even if you go just a few pence below zero.
This example also illustrates that in most cases it's better to pick an account with an interest rate charge instead of a flat daily fee, with a 5% interest rate remaining cheaper than £5 until you go more than £100 into the red, even if both were calculated daily. And unlike most flat fees, interest rate charges tend to be calculated on a monthly basis, so you'd need to stay £100 in the red for seven months to owe your provider the same £35.
Of course, you'd rather not pay your provider anything at all and keep your money. That's why it's so important to consider the overdraft when choosing a current account, especially as some providers charge both interest and a fee.