The Birth: There have been two different spider-women in the Marvel Universe to-date. The "original" Spider-Woman; to be a spider evolved into human form by the High Evolutionary, Jessica Drew made her debut in Marvel Spotlight #32.
The recent Spider-Woman, Julia Carpenter was a normal tennis player, mother, and homemaker before a mysterious group known as the Commission decided to use her in its super powers experiments. The Commission secretly injected her with spider venom in ever increasing dosages, eventually transforming her into Spider-Woman. Julia's first appearance was in Secret Wars #7.

The Legend: Jessica Drew was previously trained by the criminal organization known as HYDRA, but she rebelled when she learn of an assasination on Nick Fury. She went off on her own to make a name for herself in the super-heroine department.
Like Spidey, she displays several arachnid powers such as clinging to surfaces. Jessica can also generate "venom blasts" - bioelectric bursts that can startle, stun, or kill her prey, but they sometimes leave her depleted of energy. The general feeling of "creepiness" she exudes is the result of pheromones. Repressing these pheromones using drug therapy also repressed her powers, so Jessica decided to live with her sometimes unpredictable body chemistry. She lost her venom blast as well as her toxin immunities and other side-effects of her unique blood chemistry in Avengers #240-241, but was still hanging off of walls and even airplanes in Wolverine's early solo stories despite Dr. Henry Pym's assurances that her adhesive qualities were forever lost.
There was a time when it was believed that the villain; Viper, was Jessica's mother, but later that was all revealed as one big misunderstanding.

As for the new Spider-Woman, on her very first outing she got pulled into the Secret Wars. On returning, she joined Freedom Force at Val Coopers insistence. She fought alongside several old X-Villains, but turned against her teammates when Freedom Force captured the Avengers and placed them on trial.
After a time on the run, Julie was placed under the authority of Mike Clemson. Clemson kept her a fugitive while she worked for the government, causing her to lose custody of Rachael, her daughter, because she could not explain her absences. When the super villain group Death Web kidnapped Rachel in an attempt to force Spider-Woman to join them, Julie teamed with the Avengers West Coast and Spider-Man. Rachel was saved, but the mastermind behind Death Web, the Manipulator, induced her ex-husband to have a fatal heart attack. When the Avengers West Coast disbanded, she joined Force Works. She was seen most recently rescuing the original Spider-Woman.
Like Spidey, Julia has the porportionate strength, speed, and agility of a spider. She can cling to walls. Her most unique ability is her talent to create psiwebs, which are made of mental energy and can trap bad guys, make lines for her to swing on, or anything else she wants. The webs require intense concintration.