A business becomes insolvent when it is unable to pay its debts. (That's a simple dictionary definition, which has nothing to do with any legal process).
If the directors believe that it's possible to save the business they can apply to have the company placed into 'administration'. The administrator then runs the business, while seeking to get the company's creditors to accept only a percentage of the money owed to them and to reduce the company's outgoings (e.g. by getting rid of excess staff).
If the administrator succeeds in his task the company can apply to come out of administration, and resume trading as normal. (Several big football clubs have done that, including Leeds and Ipswich).
If an administrator can't turn a company round (or if the directors see no point in even applying for an administration order in the first place), an insolvent company will be placed into 'receivership'. The receiver will usually close the company down (although he might seek to sell any profit-making parts of the company as a going concern) and sell its assets. He will then distribute that money among the creditors.
In the UK, the term 'bankruptcy' only legally applies to individuals, and not to companies. But it's often used colloquially to refer to both 'administration' and 'receivership'. (In the USA, 'Chapter 11 bankruptcy' is roughly equivalent to our 'administration', with 'full bankruptcy' being equivalent to our 'receivership').
Chris