"It's not a cute name! The Ichi is one, and go stands for protector! It's not a cute name!"
It also sounds like 1-5, which is his age and the sign on his bedroom door. His sisters, Karin and Yuzu, are homonyms for Quince and Yuzu- Japanese equivalent of lemon and lime. They're twins. Crazy language, crazy author.
To answer your questions, Ichigo begins the anime as an ordinary student with the ability to see and speak to ghosts. His sisters can see them too, though not as well, and his father, Isshin, is continually on his back about it, claiming he feels left out.
Soul Reapers (Shinigami; death gods) are souls themselves, and cannot be seen by any human unless that human has spiritual energy (reiatsu). Ichigo's psychic abilities are a natural consequence of reiatsu- it's a very unusual attribute for a human. When Rukia is assigned a three-month shift protecting Ichigo's hometown (Karakura-cho, Tokyo), she is distracted by his ability to see her whilst she is fighting a hollow. Hollows are lost souls who have become beings of pure malevolence; they like to eat human or Shinigami souls, finding those with reiatsu particularly tasty.
Badly injured, she suggests a desperate measure to Ichigo; she attempts to transfer some of her reiatsu to him. He sucks it all up like a sponge and is able to manifest a completely badass zanpakutou (the swords used by shinigami, which are suggested to be like a mirror and extension of the soul). Her abilities don't come back after the battle, so Ichigo gets guilted into doing her job. When the three months are long past, Soul Society send two hunters out to arrest her- Byakuya and Renji, respectively her adoptive brother and childhood best friend. This is where Ichigo's powers get a little blurry. Ichigo attempts to stop the arrest, but Byakuya attacks him with the intention of destroying the physical seats of his power. Byakuya is not the sort of person who makes mistakes, yet after the battle the shady shopkeeper Urahara is able to train Ichigo to use his abilities again. Ichigo, who can be pretty dense sometimes, fails to question this. There is an explanation, but that'd be spoilers. It's a pretty freakin' surprising explanation too.
The job description of a shinigami is still something that's being implied. Seretei, the Court of Pure Souls, has a strict and old-fashioned heirachy. There's a royal family, with whom no canon character has ever come into contact. Beneath are... (this is still off the top of my head here) the Royal Guard, some other group that we also haven't seen yet, and the Gotei Thirteen (Thirteen Court Guardian Squads). There's thirteen divisions (fairly obviously) and each has a Taichou (captain) and a Fukutaichou (lieutenant/vice-captain). There are eighteen seats for officers below that rank in each division; lower ranked seats have multiple officers. The Taichou of the first division is also the Soutaichou
- the commander-general. Each division seems to have some sort of overriding characteristic- the twelfth division are deeply inquisitive, the eleventh division are excessively violent, thirteenth division are honourable to the point of fault. This could be because Taichou pick their recruits, but I think it's unusual for that to happen. When Ichimaru-taichou (silver hair, never opens his eyes) first graduated I believe he was hand-picked to enter the fifth division by Aizen-taichou (brown hair, emo glasses). Rukia is an unseated member of the thirteenth division. There are seperate jobs that seem to be associated with some divisions- the twelfth division is also the research and development institution, and the second division are the ninjas (I'm sure they have a better name than that). The Secret Special Covert Operation Corps I believe. Anyhoo, basic duties in the human world are in guarding humans from attack by hollows, and cleansing the spirits of the hollows themselves. They also help souls that never passed over to Soul Society to find peace.
Er yeah. It got sort of long-winded, but I was being very careful to be unspoilery past episode 24, unlike Wikipedia.
*~Pips~*