The youngest country in the world is (the Republic of) South Sudan, which became a nation on July 9, 2011–the 54th country in Africa–after voting to separate from the northern part of Sudan after a long war which ended in a peace agreement in 2005, after which an autonomous government was formed in South Sudan. South Sudan was at peace until December 15, 2013 when a civil war broke out along the tribal lines of the two largest tribes of the country (and involving many other smaller tribes).
The oldest modern state in Africa is Ethiopia, the only country to resist the Scramble for Africa. After resisting an Egyptian invasion a generation before, in 1896 Ethiopia resisted an Italian invasion and began to be recognized as a state by European powers. We are talking here just about modern statehood, of course, as Ethiopia, Egypt, Libya, and many other regional civilizations date back much earlier than most European civilizations, as do civilizations in Sudan such as the Kingdom of Kush (although this was more northern Sudan), Meroë and Nubia.