As you may know by now, all communications on ham radio have a callsign associated with them. With the callsign you still don't exactly know who you are talking to and where to send QSL cards to and other information about the operator.
US Callsign lookups
In the united states it is the easiest to lookup as the FCC publishes the names and addresses of all licensed hams online. Most other regions of the world do not do this. Here are some methods to access this data.
QRZ.com also known as "Que Arr Zed" or "Que Arr Zee" is an amateur radio callsign lookup site. Besides just showing names and postal addresses for operators (primarly US operators but operators can opt in to share their info on QRZ if they are not from the US), QRZ allows for operators to put a short biography and a photo on their lookup page.
Amateur Extra 7 Query tools also known as AE7Q is a site most notably known for their US Vanity callsign lookup. It shows whether or not a callsign is available for assignment and once an application has been submitted to the FCC the odds of the applicant getting the callsign they applied for.
In germany a list of all ham radio callsigns is published by the Bundesnetzagentur (Federal Network Agency) at this Website. There is also a tool where you can search for specific callsigns here.