His name is a take on the word "impossible". As of Season 4, he is the fifth and final character to use this pun (the others being, Jim, Tim, Kim and Mim Possible).
Slim Possible is a Wyoming rancher and owner of the "Lazy C" working ranch. He has a young daughter named Joss, and is the older brother to Dr. James Possible, whom he refers to as "Squirt", to the delight of the Tweebs, and thus is Kim's uncle.
Like his younger brother, Slim is an accomplished scientist and inventor. Among his creations are a series of cyber-robotic horses he called Tornado, that are able to mimic the form and function of a real horse, and an advanced surveillance satellite capable of monitoring his ranch and the surrounding area. Humble to a fault he says his brother is the real genius in the family.
Personality wise, Slim is modest, calm under pressure, and slow to anger. Despite being adept at using advanced technology, Slim seems to prefer the simple life, and to possess typical Wyoming values.
Slim Possible made his first and only appearance part in Season 3, and was introduced when the Possible Family and Ron visited him during summer vacation.
During this time, Slim helped the other members of the Possible family in foiling a scheme by Dr. Drakken to transform the world's geniuses into lackwits using brainwave altering cowboy hats.
As of the start of Season 4, Slim has not appeared again in the show, and has not been referenced by name. His picture was also notably absent from the family photo wall scene during the Season 3 finale, though his daughter's picture was displayed.
His last appearance was in "Graduation".