After the introduction, press one letter on the keyboard (either on your real keyboard or on the digital one displayed). This will show if the letter belongs in the country name and where it goes (green is for yes and red is for no). Repeat this process to reveal and knock out more letters. Once you are sure with your answer, you can press the "Guess whole word" button to guess the full country name, but, if it's wrong, you lose. If you are guessing the full name, type the country name into the text box and press Enter. If you don't want to guess the full name just yet, you can type "nevermind" into the text box and you will return to guessing one letter at a time.
Remember to Heart and Star for more updates. Hover your mouse cursor over some of the items in the room to find out facts about them, such as the Globe or Flags.