A country is any sovereign political entity. Typical factors for being a country are being independent of foreign nations, having a standing army, and having a government, but the Vatican, Taiwan, Hong Kong, and Somalia all push the boundaries of that definition.
A kingdom is a country ruled by a king. The name doesn't have any meaning, really, it's just what the people there decide to call themselves.
An empire is a country in which one nation, ethnicity, language community, historical country, etc expands to and rules over others using military force. Britain, Russia, Germany, Portugal, France, Macedonia, Persia, the Ottomans, the Mali, the Songhai, the Egyptians, the Zulu, the Swahili, the Babylonians, the Mongols, the Chinese, the Japanese, and the Americans have all at some point in history ruled over an empire. Most of them don't last very long, and they're generally not relevant in today's world stage: our present ideas about human rights for the most part preclude imperialism.