Hermione Granger (pronounced her-MY-oh-knee) is a character from J.K. Rowling's Harry Potter books. As such, any half-decent writeup on her will contain some spoliers.
Now, those who know me and have read Lisa Simpson may be unsurprised to know that I quite like the character of Hermione. And that's a useful place to start, as there are some similarities between the two, though Hermione is older, and more 'real'.
Hermione Granger is the daughter of two Muggle (non-wizard) dentists. Her parents reacted to the news that she was a witch with pride, and allowed her at age 11 to study at Hogwarts School of Witchcraft and Wizardry. She entered Hogwarts in the same year (US grade) as Harry Potter, Ron Weasley, Neville Longbottom and most of the other young protagonists of the books, and was placed in Griffindor House.
Hermione is something of a prodigy, natural talent combining with a desire to learn and an almost-obsessive perfectionism (in PoA, we learn her greatest fear is failing a test). She launches herself into her work and tends to be the first in the class to grasp a new spell or technique. The one exception to this is Divination, which she dropped in the 3rd year after finding it unlearnable. Her academic ability was so high that the Ministry of Magic authorised the use of a special device in her third year to help her study more than usual. It was unsurprising that she was made a prefect in her fifth year.
While she was originally unpopular and seemed the classic swot, squealing to the teachers and doing what she was told, Hermione seems to have flourished somewhat under the company of her best friends Harry Potter and Ron Weasley. She no longer admires authority per se, but rather justice. As such, she has poured energy into defending a hippogriff accused of attacking someone, started up an organisation to defend the rights of house-elves and got at -- and later used -- reporter Rita Skeeter. By her fifth year, she was helping to openly defy the Ministry of Magic.
Hermione has begun to emotionally mature throughout the books. In her second year, she had a crush on writer and teacher Gildroy Lockheart. During the Triwizard Tournament, she had some kind of a relationship with Viktor Krum, going to the Yule Ball with him, and a year later they are still in regular contact. However, there are strong indications of a future relationship with Ron Weasley. In her Sixth Year, she went out with Cormac McLaggan to make Ron jelous. She is also close friends with Ron's little sister Ginny, and has become somewhat less neurotic with age.
Hermione's integration into the wizarding world has been relatively smooth, though there are hints that she's had to struggle. She reads the Daily Prophet cover to cover every day, and knows more about History and politics than her class-mates, even those brought up in wizarding households. However, she misses out on speculation and rumour. She is affected by the wizards' form of racism - or possibly classism - by not being the descendant of pure-blood wizarding families, and is called a mudblood by Draco Malfoy. By the fifth year, she is also the only student in Hogwarts other than Harry Potter to pronounce the name Lord Voldemort, rather than using a euphemism.
She's been in a few scrapes over the years, from her first few weeks when she was nearly killed by a troll to the end of her sixth year when she survived a full-scale magical battle against Death Eaters. At one point, she was left with magically enlarged front teeth, which she had shrunk down a little more than strictly necessary, so she no longer needed braces. She brewed Polyjuice potion, and can produce blue waterproof fires, using an unknown incantation.
Hermione has brown, bushy hair and is not unattractive though, possibly, not strikingly beautiful. Her birthday is September 19th, and usually isn't a big deal as it's early in the term. One year, she bought herself a large cat called Crookshanks, which has functioned as her familiar at school.
In the Harry Potter films, Hermione Granger is played by Emma Watson
http://www.hp-lexicon.org/granger.html
http://www.spells-enchantments.com/abouthermione.htm
and my own recollections. I might have remembered something wrong; let me know if I have.