What are the Commonwealth countries?
Antigua and Barbuda, Australia, Bahamas, Barbados, Belize, Botswana, Canada, Grenada, Jamaica, New Zealand, Papua New Guinea, Solomon Islands, St Kitts and Nevis, St Lucia, St Vincent and the Grenadines, Tuvalu, U.K. (17 countries) are constitutional monarchies ruled by Queen Elizabeth II. Except for the U.K., the Queen delegates her powers to a Governor or a Governor-General.
Malaysia, Brunei, Lesotho, Samoa, Swaziland, Tonga (6 countries) are monarchies with other monarchs as heads of state. The Malaysian monarch is chosen by his peer monarchs in the Malaysian Federation from among themselves; the others are straight hereditary monarchies.
Bangladesh, Dominica, Fiji, India, Malta, Mauritius, Pakistan, Singapore, Trinidad and Tobago, Vanuatu (10 countries) are parliamentary republics.
Cameroon, Cyprus, Gambia, Ghana, Guyana, Kenya, Kiribati, Malawi, Maldives, Mozambique, Namibia, Nauru, Nigeria, Seychelles, Sierra Leone, South Africa, Sri Lanka, Tanzania, Uganda, Zambia, Zimbabwe (21 countries) are presidential republics. Mozambique is unusual in being a former Portuguese rather than a former British colony (though some have said it is the colony of a British colony...)
Note that Ireland, a presidential republic, is not part of the Commonwealth.