Quests are user-generated missions/adventures that players can download from http://www.asciisector.net/forum/quests/ and play within Ascii Sector. They are also a way to earn extra achievements that will show up in your Quine 5000.
After you've downloaded them into Ascii Sector's quests directory press 'Q' while landed on any planed or base to choose which quest to play. The plot is then usually kicked off by talking to a Bartender.
If you'd like to write your own story look at the commented source code of the "Kilrathi Hunt" example quest and download the Quest Maker's Manual qmanual.pdf from Ascii Sector's quest page above. (Note there is a bug when compiling quests under Linux: when including image files, like the example quest does, all *.img must be present identically, written both with small as well as capital letters, because the program is inconsistent about it, which doesn't matter under Windows, but Linux treats it as two differerent file names. So, before compiling the example quest make a copy/link for each *.img: ln holo.img HOLO.img; ln intro1.img INTRO1.img; ... and so on.)