The log is broken into three sections, where it automatically takes (interesting and sometimes cryptic) notes for you about your ongoing, pending and completed events and adventures, the outcome of some of them, and important events your character has experienced.
The Adventure Log is an outdated feature. It hasn't been updated for a long time, and many quests are no longer logged.
This section can be very helpful in certain quests. For instance, it displays the numbers given you by the Stone Faces during Daggerspire so you don't have to write them down yourself. It will also usually give you a summary of the actions you need to take to complete certain quests that involve travel.
Some adventures remain on your Ongoing Adventures tab even after completed. We don't know whether this is an indication of more adventures to come, or just bugs that need to be fixed.
Simply a list of the adventures you've completed. Some go into more detail than others. Not all adventures get logged.
The difference between an event and an adventure is sometimes not so obvious, as some events can log way more information than an adventure, and viceversa.