From the Wikipedia page 
One of the streams arose on Old Oak Common north east of the Old Acton Wells and flowed south down Old Oak Common Road and Old Oak Road. The second, known locally as The Warple ran just west of Horn Lane and south down Warple Way, named after the stream. It turned into Ravenscourt Park and fed the former moat there.
The county of London formed in 1889 was bounded by Stamford Brook and it runs along the boundary of Hammersmith and Fulham and the Borough of Hounslow under British Grove. By 1900, the brook had been covered over and is now a sewer. It runs down Old Oak Road and then Askew Road where it meets the Northern Interceptor at Goldhawk Road. It then continues down Paddenswick Road into Dalling Road (Lower) and meets the Western branch of Stamford Brook (fed by the Bollo Brook, which comes down Goldhawk Road before following King St.) The combined flow then goes past Hammersmith Town Hall before meeting the Western Interceptor just before the River Thames.
The name may be a corruption of 'stoney ford' or 'sandy ford' by which it was crossed by King Street. Stamford Brook has given its name to the surrounding area between Hammersmith and Chiswick and to local London Underground station, Stamford Brook Station. Stamford Brook has no connection with Stamford Bridge, the area of the Chelsea Football Club ground which is also situated in West London, despite the similarity of names. Stamford Bridge is built near the site of a bridge which carried the Kings Road over another tributary stream called Counter's Creek.
Plans were proposed as of 2008 to bring Stamford Brook to the surface at Hammersmith by putting the A4 underground.