A QSL card is a postcard-sized document that verifies a contact between two radio operators, typically for amateur radio. They can be used to verify an exchange between the operators for different awards, such as the ARRL DXCC Award, which is verified contact with radio ops in at least 100 different countries.
In order to be used as documentation, the QSL card has to have the sender's information (typically pre-printed on the card stock itself), and then the contact information (either hand-written or added with a printed label.) This includes the call sign of the station you contacted, the full date, time, frequency or band, and mode of contact (morse code or cw, single sideband, digital, etc.).
Most folks have QSL cards made in bulk, with stylized lettering or photos of things that show a radio operator's personality. For example, one of my QSL card designs has lots of photos of my grandkids and info about my books on one side, and the reverse side has all of my ham radio information like name, address, memberships, awards, and the above mentioned contact data. Older designs I've used were hand-drawn Calvin and Hobbes cartoons, a picture of one of my old goats when we raised them on our farm, and the USS Enterprise, NCC-1701D.
To exchange these cards, one can mail them like a postcard, pop them in an envelope with the added expense of stationary and first-class postage, or collect them in bulk and send them in to your country's QSL Buro, typically operated by the respective national radio organization. In the United States, this is the Amateur Radio Relay League, or ARRL. The last method is the slowest, but it works pretty well and is one way to build up your collection of QSL cards.
More recently, there is another option -- the digital QSL record. The ARRL has the (free) Logbook of the World, where all contacts entered by both radio operators are connected and count as a verified contact. There are some third-party ones too, like eQSL. Most folks will use the inexpensive digital version, saving their QSL cards for rare contacts.