Many nations of today once had very different names. Below is a list of some of the well-known ones.

Name                       Former Name
Bangladesh                 East Pakistan
Belarus                    Byelorussia
Belize                     British Honduras
Benin                      Dahomey
Botswana                   Bechuanaland
Burkina Faso               Upper Volta
Burundi                    Urundi
                           part of (1) German East Africa,
                                   (2) Ruanda-Urundi
Central African Republic   Ubangi-Shari
Cambodia                   Democratic Kampuchea
Columbia                   New Granada
Congo (Democratic Rep. of) Zaire, Belgian Congo
Congo (Republic of)        Middle Congo (French)
Cote d'Ivoire              Ivory Coast
Djibouti                   Affars & Issas, Tty. of;
                           French Somaliland
Equatorial Guinea          Spanish Guinea
Ethiopia                   Abyssinia
Ghana                      Gold Coast
Guinea                     part of French West Africa
Guinea-Bissau              Portugese Guinea
Guyana                     British Guiana
Indonesia                  Dutch East Indies
Iran                       Persia
Kiribati                   Gilbert Islands
Kyrgyzstan                 Kirghizia (part of Former USSR)
Lesotho                    Basutoland
Malawi                     Nyasaland
Malaysia                   Malaya, Sabah (North Borneo), and
Mali                       French Sudan
Moldova                    Moldavia (Part of Former USSR)
Myanmar                    Burma
Netherlands Antilles       Dutch East Indies
Pakistan                   West Pakistan
Rwanda                     Part of (1) German East Africa,
                                   (2) Ruanda-Urundi
Somalia                    Somaliland (Brit. & Ital.)
Sri Lanka                  Ceylon
Suriname                   Dutch Guiana
Tanzania                   Tanganyika and Zanzibar
Thailand                   Siam
Togo                       Togoland
Tuvalu                     Ellice Islands
United Arab Emirates       Trucial States, Trucial Oman
Vietnam                    Formerly North Vietnam, South Vietnam 
                             French Indo-China before that
Vanuatu                    New Hebrides
Yemen                      Part of (1) Southern Yemen
                                   (2) People's Democratic
                                       Republic of Yemen
Zambia                     Northern Rhodesia
Zimbabwe                   Southern Rhodesia, Rhodesia

Thanks Gritchka!
A few addenda to the original write-up by Schmik:

Bangladesh < East Pakistan. Before 1947, East and West Pakistan were part of India.

Vietnam < French Indo-China. Before the French took control of Vietnam, it was frequently known as Annam, which is originally a Chinese name. There are older Chinese names - there were Chinese settlements in Vietnam from the Han dynasty onward.

Iran < Persia. Persia historically was one of the major powers of the eastern Mediterranean, encompassing parts of Iraq and other lands.

Turkey was the Ottoman Empire, which stretched into southeastern Europe and much of the Middle East until World War I.

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