The is used to objects that are not immediately available. It is possible to insert anything into the Quaesitum, the best things to use are Artifacts (sometimes found in dungeons). Rare Artifacts give a better chance of obtaining something.
The interface of the Quaesitum may be slightly confusing to the uninitiated at first, but it is actually quite easy to understand:
The red slot at the top left is the slot for the primary research item. The two green slots below it are for additional items with which you can boost the research success chance. The three bars in the magenta area are bars that indicate, from top to bottom, the success chance of the current research endeavor, the chance of losing a progress point when researching a theory, and finally, the chance to lose the item that is currently being researched.
The yellow slot at the top right is for paper. As already stated earlier, you will need lots of paper since every time the Quaesitum produces a knowledge fragment, a theory or a discovery, one piece of paper is consumed. The dark blue bar is the progress bar of the Quaesitum; the harder a theory is to research, the longer it takes to complete one cycle. Finally, the light blue inventory holds the knowledge fragments, theories and discoveries the Quaesitum has produced so far.
The research mechanics are fairly simple. The more valuable the object, the better chance of success. That also applies to supplementary items. If you are using the latest recommended Technic build 7.2, you can see the research value of an item if you hover over it in your inventory. The higher this research value is, the higher the chance for a research success gets if you use it as primary research material or as a booster.
Once your Quaesitum starts researching, it will make a quill scratching noise and will start it's progress bar. What can be derived will change based on what you put in. Mundane objects will only give you Lost Knowledge. Forbidden Objects will give you Forbidden Knowledge, Tainted objects will give you Tainted Knowledge and Eldrich objects will give you Eldritch Knowledge. These knowledge fragements are the first part of the research stage of Thaumcraft. After you've gotten your first fragment, feel free to take whatever object is in the top slot out and place the fragment within.
Placing items in the booster slots is optional, but highly recommended to maximize efficiency. Unfortunately, without a Relic Hunter enchantment, it is difficult to ensure that you have enough artifacts to supply. Books make a very useful booster, so consider a very large scale paper farm to ensure you have enough books on hand.
Vis Valvesare an even better research booster than books, as they are rather cheap to craft and have an extraordinarily high research value of 15, which is three times as much as a book (5), and almost double as much as a diamond (8). You only need glass panes, some wood and some Redstone for the Vis Conduits, and some more wood and cobblestone for the levers. Efficient research boosters don't get any cheaper than this. It is a good idea to not use the vis valve to pump vis, but instead to put it in the box to be researched.
In case you are using Equivalent Exchange together with Thaumcraft, Slimeballs are a good alternative to Vis Valves or Books. All you need is one Slimeball and a Transmutation Tablet or an Alchemical Chest to get an endless supply. With their EMC Value of 25 (Books have an EMC of 96) and Research Value of 5 they are a very cheap resource.
Zombie Brain and any Vis Crystal have specific outcomes from research; you should only research these until you get Theory: Brain-In-A-Jar and Theory: Crystalline Bell. Vanilla blocks, even Cobblestone, yield Fragment of Lost Knowledge, with Book being a highly productive source of these. Forbidden Artifacts are a rare but valuable source of Fragment of Forbidden Knowledge, although Zombie Flesh, Bone and Netherrack are plentiful substitutes.
It should be noted, Researching a Knowledge Fragment of an area of research where you don't have anything left to find will yield only another knowledge fragment. Crafting yourself a Thaumonomicon will help you keep track of this.