What requirements are there for radio systems in aircraft?
What are the requirements for radio systems in aircraft?

Generally there aren't any, at least as far as "being required to have radios".

It is perfectly legal to fly around in most of the United States in a Piper Cub with no radios of any kind.
Indeed, unless it's been retrofitted with one a Piper Cub would have no electrical system at all - so you would only be able to fly under "daytime VFR conditions" in airspace not requiring communication with air traffic control (absent special arrangements).



If you elect to fly an aircraft equipped with radios (as most of us do) there are specific requirements those radios must meet, which are described in Technical Standard Orders (TSOs).  

This is not a comprehensive list, but:


For Communication radios these are outlined in AC 90-50D and TSO C-169a.  
Transponders must meet specific TSO performance criteria per FAR 91.215
Navigation Radios (VOR/ILS/ADF) must meet specific TSO performance criteria if used under instrument flight rules.
The TSO number for these escape me at the moment but I'm sure you can look them up.
GPS systems must meet TSO C145c / TSO C146c requirements (if used for IFR navigation).
The Emergency Locator Transmitter (ELT) must meet specific TSO criteria.


The requirements obviously vary based on the type of radio, and typically reference other industry standards (for example TSO C-169a for VHF communication radios sends you over to RTCA/DO-186B for the actual performance criteria).