The accrued interest on a bond is the amount of interest that has been deemed to have accumulated on the bond, but which has not yet been paid.
It is the amount of interest that would have been paid if interest was paid daily, but which has not become actually payable yet.
As a simple example, suppose a £100 bond pays 10% interest annually, so a single £10 payment is made each year. Six months after an interest payment would be exactly half way between interest payments, so the accrued interest would be half the annual payment or £5. Three months after an interest (coupon) payment, we would be a quarter of the way to the next payment, so the accrued interest would be a quarter of the payment or £2.50.
The calculation of the accrued interest on most bonds (with no special characteristics so that the amounts of all payments are known exactly in advance) is very straight forward.
accrued interest = amount of next coupon × days since last coupon payment ÷ days from last coupon to next coupon
Unless the settlement date is after the ex-div date, in which case:
accrued interest = amount of next coupon × ( days since last coupon payment ÷ days from last coupon to next coupon - 1)
The number of days will be counted using the days convention of the market the bonds trade in. The number of days is counted from the last cooupn date to the settlement date
The most common reason for calculating the accrued interest is to calculate one of the clean price or the dirty price, given the other.