Nippon is the Chinese word for Japan (derived from the Chinese characters which became Japanese Kanji). Nihon is the Japanese word for Japan (which is the way modern Japanese pronounce said characters). Though there are Japanese who still prefer to say Nippon.

The word Japan itself came from Marco Polo, who heard the Chinese read the characters for Nippon literally (Zi-pang, sun origin - which is where the slogan Land of the Rising Sun came from) with a heavy accent. Japan was so widely used by the time it was discovered to be incorrect that it just stuck and even the Japanese government doesn't care enough to issue a formal correction.