Kouga is the leader of the Wolf Demon tribe being very stubborn and headstrong. But when he meets Kagome he falls for her and changes a bit. Unfortunately this makes Inuyasha very jealous and they fight constantly over it.
Kouga uses Shikon Jewel shards to be more powerful, coincidentally making him someone the Yasha clan would meet. Kouga has been in search of Naraku along with them due to the the slaughtering of his comrades. When Naraku moved to Mount Hakurai he disturbed the Wolf Demon tribe in that region. That's why Ayame (whom he had fake betrothed to a long time ago) came back to claim her right as wife and use Kouga's strength to bring peace back in their lands. Kouga was in love with Kagome and couldn't "remember'' when he did this... and sent her packing.