Is there a database containing detailed information about airfields and runways?
The FAA publishes the NASR database. This is an official, authoritative data set that covers all airports, airways, routes and navigation aids under FAA jurisdiction. The data files are not human-readable though and require special software to display.
Other civil aviation authorities in other countries have similar products. Some are available for free, some are not. Many countries claim intellectual property rights on their data and it can be quite difficult to get at/use it.
Some companies like Jeppesen do this collection for you: they have agreements with aviation authorities, obtain each authority's data sets (like the FAA's NASR linked above), and compile all of them into a single large database. This database is then reprocessed in forms usable by avionics like GPS receivers or Flight Management Systems (FMS). These countries charge a significant fee for this service, and there are licensing terms that limit your use of the data. The fee is substantial depending on the format you get the data in (up to thousands of dollars or more for FMS systems for large transport aircraft) and puts the dataset out of reach of a casual user.
The U.S. Department of Defense publishes the DAFIF which is a similar compilation of worldwide data. This database used to be available publicly for free, but is now restricted to Department of Defense personnel (it is not classified but it is not available to the public).
There are many websites based on one or more of the datasets above, but as noted in comments, these websites are not official or authoritative. They also generally have their own license terms that limit the way you can use the information and that prevent you from downloading the database in bulk. This all stems from the intellectual property claims made by many civil aviation authorities. It costs countries a lot of money to maintain the data and they want to be able to generate some revenue from it.
So the bottom line is that for some regions like the United States, official data is readily and freely available; for other regions, not so much. (And among developed countries, the U.S. is the exception; most do not offer their data freely.) To get data for these other regions, or to create a global dataset, would involve a lot of effort and money (beyond the reach of a normal individual). Companies that do this work for you charge hefty fees to provide the data to you.