The company, in its four or five forms, is a special case of the body corporate. Historically, there seem to have been four different types of corporation:
The chartered corporation is the best starting point: before the prerogative power was brought under control in the seventeenth century, the King (or indeed Queen) would issue charters to groups of individuals, requiring the courts and others to treat the group as though it were a single body. "corporation" basically means turning something into a "body", from Latin "corpus", which means the same thing. You didn't want to have a situation where someone sued the Bishop of Winchester, but he died before the lawsuit was resolved, and the new Bishop was allowed to walk away from the lawsuit, and this principle was generalised to groups of people who existed at the same time, rather than who succeeded each other over time like occupants of a bishopric.
Prescriptive corporations were a legal fiction: it was absurd to treat groups of people who behaved as though they were a single organisation in a way which pretended that they were NOT a single organisation. You didn't want a lawsuit collapsing because, say, Oxford University's charter was so old that no-one could still find it. Maybe it never had one - it had just always operated as a single entity, and it would be ridiculous for it suddenly to be treated as a collection of hundreds of individuals all parties to some lawsuit that had just got jarringly more complex. So prescriptive corporations are organisations so old and so cohesive that it's fair to assume that they probably were once given a charter, but it was too long ago to matter. A sensible legal fiction.
Statutory corporations are corporate bodies set up by Parliament, after the Crown was brought to heel in 1689. Each one had its own act of Parliament, just as each chartered corporation had its own Charter. The main difference was that the document issued from Parliament rather than the Crown (this has some implications in the present day for whether a corporation can make rules which are binding on people who aren't members: important for harbourmasters, universities and so on). You can still dig up individual acts of Parliament for the Hudson Bay Company, or the Bank of England, and so on. When Halifax Building Society merged with the Bank of Scotland a few decades ago, this required an act of Parliament too.
Registered corporations are what we now call companies. One act of Parliament, since roughly the 1850s, has allowed a government official to create new companies. The current law on this is the Companies Act 2006.
The miscellaneous other bodies created by act of Parliament, e.g., county councils, tend to be statutory corporations.