There is a meeting of the Privy Council today, as many newspapers have noted, although their main reason for covering that piece of news is to point out that Jeremy Corbyn, the newly appointed leader of the Labour Party, won’t be attending (nor will over 600 other Privy Counsellors, mind you – there are usually only three or four Privy Council members present at meetings). It is clear that Corbyn has decided to accept the Queen’s invitation to join the Privy Council, as Parliament's website describes him as the Right Honourable Jeremy Corbyn, the term Right Honourable denoting a Privy Council member.
The Privy Council dates back to the times of the Norman monarchs, when it consisted of people appointed by the monarchs to advise them on matters of state, similar to the role of today’s Cabinet. Today’s Privy Council has a more limited role (see this article and the Privy Council’s website).
Privy has various meanings, but the Privy of Privy Council means, according to the OED, “Belonging to or reserved for the exclusive use of a particular person or group of people”. This is an obsolete sense of the word and remains only in set expressions eg Privy Purse (the personal income of the sovereign, which comes mostly from the income of the Duchy of Lancaster) and privy seal (the monarch’s personal seal).
The spelling these days is Council, but it was originally a variation of Counsel, to indicate the advisory nature of the body. The s in the spelling remains for the members of the Privy Council; they are generally called Privy Counsellors. The official website says that both Privy Counsellor and Privy Councillor are correct spellings, but the former is preferred.