It began officially in February 1870. It had begun on an ad hoc basis in the late 1700s.
HISTORY OF THE NATIONAL WEATHER SERVICE
The National Weather Service has its beginnings in the early history of the United States. Weather has always been important to the citizenry of this country, and this was especially true during the 17th and 18th centuries. The beginning of the National Weather Service we know today started on February 9th, 1870, when President Ulysses S. Grant signed a joint resolution of Congress authorizing the Secretary of War to establish a national weather service. This resolution required the Secretary of War "to provide for taking meteorological observations at the military stations in the interior of the continent and at other points in the States and Territories...and for giving notice on the northern (Great) Lakes and on the seacoast by magnetic telegraph and marine signals, of the approach and force of storms".
https://www.weather.gov/timeline
FROM PETROGLYPHS TO A NATIONAL CLIMATIC DATA CENTER
People have observed and recorded the weather for thousands of years. Native American petroglyphs often depict rain, the sun, or lightning. Much later, colonists from Europe began recording journal entries about the weather and natural environment they observed. By the late 1700s, accurate weather instruments, such as thermometers, were available to professional and amateur scientists. Historical figures, such as Thomas Jefferson, routinely recorded daily weather conditions. For the next two centuries, these weather observations accumulated in archives scattered across the country. In 1951, the Federal Government moved all weather records to Asheville, North Carolina, where the archives at the U.S. Weather Bureau, Air Force, and Navy combined to form the National Weather Records Center (NWRC).
http://www.ncdc.noaa.gov/about-ncdc
