Mini-Dungeons are solo instances you find in the world that are usually closed off until something happens in the area, such as events. Ab easy one to try out is in Queensdale at Beggars Burrow, North of Lake delavan. If you start at Ojon's Lumbermill Waypoint, head north across the lake and you will find an entrance guarded by bandits. Kill them, go inside, and fight your way through to a Splendid Chest and an achievement! not too tough! There is another entrance at the graveyard, but it is only accessible during Storyline events and they don't count towards the achievement. :(
Below is a link to the mini-dungeon wiki with a list of all of them! Good luck and enjoy!
Wiki with List of Mini-Dungeons