A Destructive Analyzer is a machine that deconstructs components to reverse-engineer their technologies. Deconstruction itself is easy. Simply stick items into the destructive analyzer and turn it on. It is possible to eject items from the deconstructive analyzer if, say, security notices their missing laser guns. As always, experiment! Some technology on the station is just so basic that the analyzer won't even bother with it, but things like bruise packs, flashbangs, and some of the odds and ends in your very lab will all work.
For related devices see:
List of Destructible Items
Conventionally you will use the Protolathe and Circuit Imprinter to construct the items you are going to be breaking down. Breaking down an item will raise the tech levels on the system to that of the item that was broken down, however it will instead raise the tech levels on the system, that the item possess, if the systems levels are the same as the items levels. For example, the system has a research level for power of 3, you break down an advanced capacitor, which also has a power level of 3, the systems research power level raises to 4.
If you need assistance or would simply like to quickly find out what items you can break down to progress your tech, try the following tool. Kyet's R&D Tool / R&D Item List
To use the tool, simply input the tech levels the research terminal already possess and a list of potential researchable items will be present under the table. Applying any of the items will update the tables tech levels, allowing you to rapidly use it to progress your research efforts.
Should you, for whatever reason, need to find items for specific tech levels, you can press filter next that specific tech on the table, which will cause the potential items to break down to only those for that specific tech.