A charity is any organisation set up for the purposes of promoting the public good. This may be (and almost always is) the good of a sub-section of the public but you can't set up a charity to benefit your family. In 2006 the idea of public benefit was tightened up and you can see here what the Charity Commission says about it.
An organisation has a form and a status. The form for most charities is either an unincorporated association or a limited company. There are also trusts, industrial and provident societies, charitable incorporated organisations and limited liability partnerships. See Get Legal for more information about these.
As an unincorporated association the organisation is not a legal entity and cannot own anything or employ anyone - all this is done by the trustees. As a company the organisation is itself a legal entity - so for example the company can be awarded a contract.
The status of an organisation is about whether or not it is a registered charity. An organisation does not have to be registered to be a charity - but it helps in particular for organisations which use public money, either individual donations or grants. Having charitable status infers a degree of protection for trustees if they abide by charity law and provides a framework for operating the charity. It also imposes responsibilities on trustees and required reporting to the Charity Commission.
Therefore, any organisation which is deemed a charity will have a legal form and may or may not have a registered status.