Niger is a country in West Africa which is named after the Niger River, which flows west of the country. The origin of the river's name, however, is uncertain. It is purported that its name derives from the Tuareg name n'eghirren, which means "flowing river".
Because of French colonization, the official language of Niger is French. It follows that the correct pronunciation of Niger is in French, and should also be followed by English speakers. The phonetic pronunciation of Niger can be transcribed as Nee-szehr. First of all, the first syllable "Ni" contains an "ee" sound for the "i". The second syllable contains an "sz" sound that equates to the "s" in the word "vision", or even simpler, it is similar to a "j" sound. The following "e" is an open "e". The final "r" is a gargled French "r", but for the non-native it isn't fundamental for a good pronunciation.
A note on the proper way to call the population of Niger: the population of Niger are Nigerien, while people from Nigeria are Nigerian.