Is there an atmospheric data database somewhere on the web?
By your edit, it seems you are more interested in the "big data" side of the science. Model reanalyses contain large amounts of historical weather data, so that is a safe bet for ample amounts of data. NOMADS, which is a service provided by the National Weather Service is a good resource for model data. There is another version of NOMADS for operational data; I'd recommend that. For generic data, RDA has data. NASA has a server that also has a lot of data.

You can also generate your own weather data for prediction and the like, by using models like WRF or SPEEDY. 

Departing from model data, you have observations, you can access ground-based observational datasets via NCDC. NCDC also has satellite data, radar data, and so on down the line (you don't need me to explore the site for you).

Other countries have other datasets, though many come with a price. ECMWF, for example, has reanalysis products, operational datasets, etc. The same is true for JMA, and other countries with their own weather service.