While the Prime Minister is selected by the President, all other ministers are appointed by him on the advice of the Prime Minister. While selecting a Prime Minister, the President is restricted to the leader of the majority party at the Centre or the person who is in a position to form a government and prove his majority later on. The allocation of portfolios to the Ministers is also done by the President as per the Prime Ministerial advice.
The Prime Minister is at the head of the Council of Ministers and enjoys a "First Among Equals" (primus inter peres) status. The Council cannot continue to exist in the event of resignation or death of the Prime Minister.
The term Council of Ministers refers to all the Ministers, whether Cabinet, State or Deputy Ministers. In reality, there is no classification of ministers in our Constitution. All this done has been following the British practice in this regard.
The Council of Ministers is collectively responsible to the Lok Sabha (Art 75). Theoretically, if a no-trust motion is passed against even of the ministers in the Parliament, the whole Council is supposed to resign as a consequence. (which has not happened so far)
The entire Council of Ministers seldom meets as a single body. It is the Cabinet, an inner group within the Council, which takes all major decisions and which shapes the government policy. While Cabinet Ministers can attend all Cabinet meetings as a matter of right, the Deputy Ministers and Ministers of State can come to the meeting only if they are invited.
Ministers may be chosen from either House of Parliament and a minister, who is a member of one house, has a right to speak and participate in the proceedings of the other House, but he cannot vote there. A person who is not a member of either, can be appointed a minister, but he must get elected to either House of Parliament within 6 months of his appointment.