IGMs (In-Game Messages) are delivered by pigeons to and from players in the game.
In the lower left corner of the game screen, click the first of the five circles marked "messages". Here you can read messages you have received or that you have sent, write new messages to other people, and read game report messages. Gold club members can also move messages they want to save to an archive section, and have a notepad for writing themselves reminder notes.

When you start to type in the "Recipient" field, a list will appear of possible matches to help you find the exact player name you want.
In the top right corner of a blank message you will notice the word "massmail". Click this and you will get a list of any clubs you have sufficient privileges in to send massmails to (or your guild). You can also send a message to your entire friend list using this function. This costs 2 diamonds per massmail (not per recipient), but it's free for Gold Club Members.

When you reach gen 2 you will be able to meet Penpalus, the owner of a stationery store situated east of the arena. You will be able to buy special paper with which you can personalize your IGMs. Special paper gives a choice of attractive backgrounds to your IGMs which add a characteristic tone to your correspondence. The sheets are offered in packets of 5 units, and you can buy and hold a maximum of 100 sheets at once. The prices vary, depending on your level. The higher the level, the more expensive the sheets become.

Tip: If you want to send a massmail with special paper, you need one sheet for each person receiving it so everyone can see the paper. If you send to more people than you have paper for, some will receive regular IGM backgrounds.