A Gaiden chapter (also known as a Sidequest, particularly in the Japanese version of Fire Emblem: Awakening and Paralogue in the English version of Fire Emblem: Awakening) is generally an optional Chapter that is played when the player has met certain conditions. Gaiden chapters were introduced in Fire Emblem: Thracia 776, and have appeared in Binding Blade, Rekka no Ken, The Sacred Stones, Shadow Dragon, and Shin Monshō no Nazo after that. They often feature Fog of War, and contain recruitable characters that are normally unobtainable. In chapter listings Gaiden chapters are listed as the number of the chapter before it, but with an "x" after the number. Chapter 19xx of Fire Emblem Rekka no Ken is unique, in that it is the only sidequest of another sidequest. A Gaiden Chapter can be quite hard to unlock sometimes. Major exceptions are Chapter 29x/31x of Rekka no Ken and Chapter 5x of The Sacred Stones, which are accessed automatically.
They are not part of the main story, but unlocking Gaiden chapters may impact the ending. For example, in Fire Emblem: Binding Blade, unlocking all Gaiden chapters is required to access the "true" ending of the game (since the player needs to obtain all legendary weapons and have them intact while facing King Zephiel in order to unlock a few extra chapters that will lead to the real final boss, Idenn). However, in Fire Emblem: Shadow Dragon, Gaiden chapters are chapters that can only be accessed by having fifteen units or less (in the whole army). These chapters contain recruitable units that are quite useful (like Athena or Horace). However, there are many characters that will eventually become worthless in the end, so if the player feels that they want to go to these chapters, they can sacrifice these units.