Back office functions are support functions such as clearing and settlement, accounts and compliance. Back office work is vital but generally regarded as less interesting and desirable than front office, or even middle office jobs - it is not difficult to find articles advising people on how to move from the back office to the front office, but no one ever wants to move the other way.
Because it is easier to define, and individuals have less discretion in how to carry out their work, the back office functions are fairly easy to out-source.
It is considered good practice to keep the management of front office and back office functions well separated, so that independent monitoring of, and controls on, the actions of the front office exist. One of the best examples of why this is important comes from the collapse of Barings Bank, which occurred partly because the notorious Nick Leeson was in charge of both trading and the back office.