Circuits are used to construct Bat-Packs, Jetpacks, Lapotron Crystals, HV Transformers, Advanced Circuits, Miners, Pumps, Canning Machines, Compressors, Electro Furnaces, Extractors, Macerators, Electrolyzers, Personal Safes, TFBPs, and Tesla Coils. They serve no other purpose.
See: Crafting Guide
As well, a new and surely important part of advanced machinery are these new Electric Circuits, officially labelled EEEAAAUUU Multi-Functional-All-Around-Ultimate-Usage-Circuit.
If you want to craft your own machines, don't forget to build some circuits first!
This little thingys are highly advanced technology and can be used for any kind of electric control system! Anything past a basic Furnace can't work without them.