Medals are awards you can earn through completing certain secret tasks. Tasks range in difficulty from sleeping to winning 500 duels, with each task having a maximum of three levels - gold, silver or bronze. Tasks with only one level have a wreath medal instead of a gold medal.
Once you complete a task which has a medal assigned to it, you will receive a report stating which medal you received. You will not be told what you got it for, but the name, graphic and time stamp will help you figure it out yourself. A medal report looks like this:
You can view your medals by going to the character menu and clicking on your profile picture. Here your medals are listed chronologically in the order you earned them. If you hover over a medal icon then the full details of that medal will be shown. The bar along the top shows the four medals that listed by default on your profile (others can still view all your medals by clicking on your profile picture there). You can drag any four medals up into the top row to be displayed.
You can view other people's medals in the same way. Simply go to their profile to see their top row of medals, or click on their profile picture for a complete list. Hovering over a medal icon will show the full details of that medal, same as your own profile. You can also view your own medals from your profile, however you will not be able to edit the top row.
A list of all known medals can be found here.
Although medals requirements, many users have been attempting to compile accurate records of all known medals and how to obtain them. These records are not complete and necessarily accurate, except they act as a good guide to a user interested in obtaining them.