Mini-Quests are tasks for players to complete. They were added in Beta 20, aka Labor Day 2014.
Next time you walk up to an NPC that you're friendly or neutral with, talk to him/her. He/she may offer you a quest to complete. When that happens, a panel pops up on the screen that gives you the choice to either accept or decline the quest. The quest can consist of any number of tasks, from killing a certain number of enemy units to simply getting a Dwarf a few raw porkchops so the family can enjoy supper.
Once you have done your deed, return to the NPC who offered you the quest (making a custom Fast Travel waypoint near them can help with this). They will take the items you gathered for them (if applicable) and give you some reward in return -- most often Alignment with their faction, the exact amount depending, of course, on how difficult the quest was to complete.
In addition, any NPCs who you are currently questing for will have a Red Book icon floating above their heads, making them easier to locate.
To keep track of your quests, it is recommended that you use the Red Book. If you spawned in a single-player world, you will find a Red Book in the starter chest, but if you started in multiplayer, the Red Book can also be crafted -- see the article about it for all the details.
A notable feature about questing is that it allows you to get alignment without killing units.
Quests are still a WIP and more epic ones (with greater danger and reward) will be added in the future.