Side quests are additional tasks or sections of the game that are not required to be completed to finish the game but can be done to gain additional abilities or equipment. Often the pay off for completing these side quests is fairly substantial but the tasks or duties can be fairly challenging. In Final Fantasy IV there are really only two types of side quests that you can do: find and fight strong monsters to gain as summons for Rydia or find a series of items and people to get special weapons or armor.
The additional summonings that Rydia can get are fairly powerful and useful and the walkthrough does use some of these items in the final videos. The same goes for the equipment. Below is the list of summons and a brief description of their attacks:
There are only two direct side quests for equipment in the game, one is an upgrade to Cecil's Light Sword. This is the only sword in the game that you cannot throw or sell, denoted by the gray text color in the menus. The reason is that you can upgrade this to a more powerful version called Excalibur later in the game. The other side quest comes very late in the game but yields you the most powerful body armor in the game and is wearable by anyone.