The First Key is the first Key To The Kingdom. When the Architecht left, she left her Will, and The Will picked Mister Monday to be the Trustee of The First Key, although he wasn't known as that yet. When the Will was divided and the Trustees missused their powers, the First Key was kept in its seperate parts, the Lesser and Greater. Mister Monday held the pieces of the Key in his sleeves for easy accses presumably.
As a part of the Wills plan to return order to The House, and Arthur lay on his death bed, The will made Monday give the lesser part to Arthur. Since then, Arthur has obtaineed the Greater Part and after that, he left it under the Stewardship of Dame Primus.