Inu-Yasha is the son of a demon father (Sugimi) and a human mother (common rumor: Lotus) making him a hanyou: a half-demon. He's referred to as a "half-breed" by his half-brother (Sesshomaru) and many other demons. Inu lives in Sengoku Jidai (Warring States Period) of Japan, which is in the 15th century just about.
50 years prior, Inu lived in a small village that had a high priestess named Kikyo. She protected a mysterious jewel called the Shikon no Tama (Jewel of Four Souls) which could granta demon strong powers. Inu desired the jewel to become a youkai (demon). But, as time went on, Inu attacked the shrine less and less, for fear of hurting Kikyo, since he loved her.
A hanyou called Naraku alsp wanted to jewel, and he wanted Kikyo's heart. He hated Inu, so he formed a plan and tricked Inu and Kikyo into killing each other. But the Shikon Jewel (nicknamed "sacred jewel") was destroyed, meaing Naraku couldn't take it. EVIL MAN YEAH!!!! okay no more randomness.
Inu-Yasha is a white dog demon: even though he's a hanyou, he has long white hair, fangs, claws, yellow eyes, a super nose, and little dog ears. In one episode (episode 36) Inu-Yasha gets really pissed off from Kouga calling him a "mutt" and fumes over his dog-like abilities.
Even though he still loves Kikyo (although I don't know why) he also is starting to feel differently about Higurashi Kagome (Kagome Higurashi) who is supposed to be Kikyo's reincarnation. (reincarnation: same soul as someone who lived before, and look-alikes bodywise.) He even KISSES her, goddamit!
Inu's favorite weapon is the sword left behind by his father for him. The sword was created by Toutousai (TOE-toe-sigh) a dog demon as well, although a wierd one. The sword is called Tetsusaiga (Iron Pulverizing Fang) and has many magical powers. Inu is just staring to unlock all it's secrets. Sesshomaru wants Tetsusaiga dearly, since the sword left behind for him (Tenseiga) cannot cut. Inu's father had both swords made so his two (and only) sons were unable to fight each other. More about Fluffy and Tenseiga later.