The words Nihon and Nippon are written the same in kanji, they are combined from the kanji NICHI (sun) and HON (origin), hence the name "Land of the rising sun". You can find a nice explanation here: Japan FAQts.
When combining these kanji the combination of these two kanji results in a contraction: NICHI+HON = NIPPON
as a reference: one pencil is counted as ICHI+HON = IPPON
This has been the official name for a long time, however, as happens in all languages, the word has evolved into NIHON, which is now the official name.
NIPPON is still used, but has an archaic ring to it nowadays.