A Prime Minister is the head of government and, in some countries, the only person who can consult with the monarch or governor general, advise them to dissolve or prorogue parliament and call a snap election. It is a role that carries enormous political power and prestige, but one that also has its limits. Various conventions and political reality constrain the power of premiers, as they must often reconcile competing factions within their own parties and Cabinets. In 1976 Harold Wilson reflected that only in wartime could Prime Ministers, “dictate to their Cabinets.”
A PM can be appointed from any party that commands a majority of seats in parliamentary elections, though they usually come from the largest party in the coalition government formed after the election. They may be sworn in at a formal meeting with the monarch, or in an audience with the Monarch. This can take place either physically or virtually, but if they are sworn in in an audience, the Monarch must have the requisite confidence to ensure the appointment will be honoured.
When a PM is appointed, the Monarch asks them to form an administration. They must then agree to do so, and this is the point when they officially become Prime Minister. At this time the Monarch shakes hands or bows, and they are shown out by an equerry. They may then immediately begin their work, or wait until they have been formally sworn in in a Privy Council meeting, or at a subsequent session.