The National Oceanic and Atmospheric Administration (NOAA) is actually a relatively extensive US government agency which has many different functions (not all of which are weather-related). As new issues like global climate change have come up over the years, so has NOAA expanded its services since its creation in 1971. Currently, the bureaucracy contains of several major divisions (each one its own agency), while NOAA itself is a division of the US Department of Commerce.
For most people in the US, the weather service is perhaps the most eminent part of NOAA, as it is in charge of analysing and making predictions about current and immediately-upcoming weather patterns; based on data recieved from other agencies. The meteorologists from the National Weather Service are those who come up with the weather forecasts.
The work of the ocean service relates to management of the US oceans and coastal areas (such as beaches). Duties of the agency include keeping the nation's coasts healthy & clean, as well placing things like buoys, channel markers, and lighthouses in areas that need them. The NOS must also evaluate and protect coastal areas from any potential perils of the sea - tsunamis, earthquakes, beach erosion...and the like. The ocean service also conducts a broad range of ocean-related scientific research.
NESDIS, as the agency is referred to, operates and manages the US' environmental satellites - which do things like collecting weather/climate data and whatnot. Says the agency, these things are necesary "to provide and ensure timely access to global environmental data from satellites and other sources to promote, protect, and enhance the Nation's economy, security, environment, and quality of life." Presumably (although I'm not certain of this), the National Weather Service gets most of its data from NESDIS.
There are two main functions associated with NMFS. The agency must manage and ensure the success of US-based fisheries; but it is also in charge of tracking and protecting marine animal and plant species. Because of this, NMFS usually has both the environmentalists and the fishermen pissed off at it.
See National Marine Fisheries Service
"Where science comes to life" -- so says NOAA, anyway. OAR is the more abstract, long-term oriented division of NOAA in that its focus is on studying and gaining scientific insight into the environmental (atmospheric and oceanic, in this context) phenomena present in our world. The agency's Space Environment Center in fact studies environmental phenomena that aren't present in our world.
Other NOAA-related things
NOAA is responsible for researching and addressing issues related big/long-term climate phenomena such as: global warming, el niño, la niña, droughts, flooding, et cetera.
Another of the agency's functions relates to the management of the coastal zones, which will sometimes have unique environment/habitat conditions.
Charting and Navigation
NOAA also maps the ocean/ocean floor, produces nautical charts, and tries to deal with navigation-related issues.