In the Doctor Who universe, each incarnation of the Time Lord known as "the Doctor" has a different appearance, personality and style. If you ask anyone familiar with the series to describe the Fourth Doctor, he or she will inevitably mention the scarf.
Throughout his run, Tom Baker's Doctor sported an obscenely long multicoloured scarf. The story goes that, after being inspired by a painting of (a scarf-clad) Aristide Bruant by Henri de Toulouse-Lautrec, the show's costume designer commissioned a scarf from a friend and provided her with a large amount of yarn. She then used all of it.
The original scarf consisted of stripes in brown, beige, red, yellow, green, grey and purple. As the series progressed, the scarf became worn out in parts and new sections — featuring pale blue stripes — had to be added. In his final series, the Fourth Doctor sported an altogether different scarf made of burgundy and orange yarn.
Knitting your own
The BBC once released an official pattern. In more recent times, numerous websites have been devoted to patterns for the scarves that appeared in various episodes.
The scarf is knit entirely in garter stitch, so all one needs in order to make one is knowledge of the knit stitch, yarn in the right colours (the knitting sites listed in the reference section have excellent colour suggestions) and patience. Some enterprising people make them to sell.
wittylittleknitter.com (these two devoted entirely to the folklore and knitting of the scarf)
Fourth Doctor: Appearance on Wikipedia
254 words, not including the Quest name and this disclaimer.
I also would have preferred to have noded this under Doctor Who scarf, but this already existed. Perhaps a lovely and helpful staff member can retitle this node for the sake of geekular accuracy. Many thanks to Aerobe for retitling the node according to accepted Whovian conventions!