The name Inu Yasha is composed of two separate words: Inu, dog, & Yasha which means either 'demon' or 'female demon' leading to a great deal of buzzing in the IY fan community.
As for our favorite character,  Inuyasha's life before the series remains vague, and yet not so vague.  Thanks to the 3rd movie, Tanka Hadou no Ken, we know that his father died when he was still a newborn.  His mother, a woman of nobility, had died but was revived by Tenseiga, the sword later willed to Sesshoumaru.
His mother dies at some point during his life before we are introduced to him, at the age of 15.  This has been stated by Takahashi as Inuyasha's age, though in apperance most believe him to be 17.   He meets Kikyou, a young woman who serves as the protector of the Shikon no Tama, a jewel of emense power.  
Through trickery beyond their control, they are lead to believe that the other betrayed their trust.  Kikyou sealed Inuyasha to a tree, and then died of injuries she believed Inuyasha inflicted upon her.
50 years later, Inuyasha was released from his spellbound imprisonment by...
Kagome is a frank fifteen year old girl whose only interest is studying to pass the exams.   Her family is made up of her grandfather, mother and younger brother Sota. Her home is an old shintoist temple where there's a Goshinboku (a tree planted inside of a temple which is worshipped as a sort of guardian angel) which has more than 500 years and an eloquent secret well.   One day she goes looking for her cat Buyo in the building which surrounds the well but is attacked by a centipede monster who drags her into it and into a late period of Japanese history, an era of bloody battles between lords and of magic and mystery and where her arrival gives way to a chain-reaction.   Kaede believes her to be her dead sister's Kikyo reincarnation because not only does Kagome look like her and has her mysterious powers, but also because the Shikon no Tama was guarded inside her body
The Characters
A great many thanks to ChibiRuby for the pictures on her site.  All the images with the names in kanji were taken from her site. http://www.chibiexplosion.com/inuyasha/index.html
The Shard Hunters
Miroku is a young Buddhist monk, with a family curse.  His grandfather fought and was defeated by Naraku and as such, his grandfather, and all his male descendents, including Miroku, are cursed with a black hole in their right palm.  After a certain amount of time, this black hole consumes its owner, and thusly Miroku is fighting against time to defeat Naraku and stop the curse from consuming him as well.  Miroku has another curse, lechery.  He is infamous for asking lovely young woman to "bear his child", almost as often as he is groping said young woman.  But he has met his match, for he has recently proposed to the beautiful, but spirited ...
Sango is a demon exterminator, trained by her father all her life.  During one mission, they were tricked by Naraku and Sango watched her young brother Kohaku murder their father and their companions.  Nearly dying herself, she was saved by Naraku to use against Inuyasha, but she freed herself and now travels with the group in the hopes of avenging the murder of her villagers, and rescuing Kohaku from Naraku's control.  
Shippou is a kitsune (fox demon) orphaned at the beginning of the series.  He encounters Inuyasha and Kagome in an attempt to steal the Shikon no kakera to make himself stronger and avenge his father's murder.  After the task is complete, he joins Inuyasha and Kagome in their travels.  While he is not strong in the fight, he does help out as much as he can, using his kitsune tricks at his disposal.  He also provides much comic relief, teasing and taunting Inuyasha to no end.
Kirara is Sango's pet and companion.  She is a fire-cat, and very little is really known about her.  She has two forms, the small cute cat sized form, but when needed, she can transform into a form the size of a small horse.  It has been hinted in the series, that Kirara is very old.  In a small bit at the end of the episode about the origin of the Shikon no Tama, you see a full grown Kirara standing at the side of Midoriko, the woman who created the jewel.
The people in their lives
The Bad Guys
Kaede is the priestess in the village near Inuyasha's Forest.  She is the younger sister of Kikyou, and was there the day that Inuyasha was sealed and Kikyou died.  She has protected her village from that day forth, and when Kagome released him from the seal, Kaede was the one who supplied her with the subduing rosary, and sent them on their mission to retrieve the pieces of Shikon no Tama.  Kaede has a light banter to her mannerisms that comes with the benefit of old age, and the appreciation of the power of youthful attitude.
Kikyou was the original protector of the Shikon no Tama, a young woman that most believe to be is 17 years old,  She is a stoic powerful figure in the flashbacks, and a cold silent figure in her 'undead' life.  Most want to hate her, but truely she is a character to be pitied.  Dying with a lie in her mind and heart, and having to live another seperated life from those around her, driven by mysterious that many have tried to decipher.  Perhaps she will be able to finally achieve the goal she walks towards, episode by episode.
Kouga is the leader of a pack of wolves that Inuyasha nd the group encounter during their travels.  He discovers that Kagome can see the shards and kidnaps her, then "falls in love with her".  He constatly comes across the group, reminding both kagome and Inuyasha of his claim that Kagome is "his" woman.  However, he already has been promised to a girl, even if he doesn't admit to remembering his promise to a young girl under the rainbow of the moon.
Ayame is the granddaughter of a wolf pack leader of the mountains.  As a young child she was rescued by Kouga who promised the little girl that when she was grown, and come down from the mountiain, he would make her his wife.  It was this promise that gave the young girl strength to grow, and the hope for that one day.  However when she finally does meet him again, he doesn't remember the promise, or so he claims.  Ayame leaves again, but with an air that promises she hasn't given up on Kouga just yet.