In the case of the Gempei War, the two sides listed on Wikipedia were led by Minamoto no Yoritomo and Minamoto no Yoshinaka. These men were cousins, having a shared grandfather in Tameyoshi no Minamoto. All three were members of the Seiwa Genji, descended from Emperor Seiwa. It's worth noting that there were 21(!) different branches/lines* of the Minamoto (Genji), each descended from a different emperor. The Seiwa branch is the Genji line that produced the primary figures on the Minamoto side in the conflict.

Basically, these men were part of the same branch of the same clan, but ended up at odds because Yoritomo's brother Yoshihira killed Yoshinaka's father and seized their lands. Yoshinaka didn't want to accept Yoritomo's leadership and seized Kyoto on his own, but then lost the reigning emperor's favor and was eventually killed by his cousin Yoshitsune.

*Some of these families were extremely small, depending on the number of descendants of the given emperor.