"Olden days" is a fairly relative phrase.  Some younger students might consider olden days any time before Instagram existed.  They can't fathom a world that existed before cell phones and texting, so I'll use that as my "olden days" line in the sand. The following is a list of ways to send messages to another person with pre-cellular technology.

• Talking.  Verbal communication between people has been used for thousands of years. 
• Native Americans used smoke signals, drum beats, and torches in key positions. Interestingly, this torch/lantern concept was still in use during the American Revolutionary War.  That's how Revere was told if the British were coming by land or sea. 
• Tap codes.  Prisoners of war or people in prison can communicate by "tapping" either the metal bars or the walls inside the cell.  This method is similar to Morse Code. 
• Telegraph communication and Morse Code. 
• Body language. 
• Written letters.  Letter writing has been in use for a long time, but the method of transport has changed and adapted to the situation and the available technology.  The postal service still currently serves this purpose; however, carrier pigeons have been used as well.   
• A runner.  The Ancient Greeks would give information to a runner who would run the information to somebody else.  This is where we get our modern marathon distance from.  Phidippides was given the job to run 26 miles to Athens to give some key information regarding a battle. 
• Radio communication.  This type of communication could be mass media radio communication, but it could also be walkie-talkie, CB radio, or Ham radio communication. 
• Cave paintings. 
• Telephone through wired connections.  This was first invented in 1876 by Alexander Graham Bell.  

By "olden days" I will assume you mean the days before technology, or perhaps even the days before the telephone was in widespread use. Letters were a common means of communication prior to the electronic age. Unlike today, when letters take as many as two to three days or longer to travel across the continental United States by mail, letters written during the Victorian era could be exchanged on a much faster basis. Generally the wealthy classes had servants delivering letters to local residents by carriage or horseback; and in this way an urgent correspondence could take place and several letters a day could be sent and received. The "Pony Express" was a method used to deliver mail in the western states after the Civil War ended and settlements sprung up slowly with the building of the railroads. Horses were used and could cover an amazing amount of territory in a short period of time, making it a very efficient message delivery system.

The invention of the telegraph in the 1830s did allow a sort of electronic message to be sent, but this was mainly for military usage. However, the telegraph could be used to send an urgent message for personal delivery called a telegram. When a soldier was killed in war, as during World War I or World War II, the family would often be notified by a telegram sent by the US government.