A perfect conductor reflects all frequencies of electromagnetic waves because the free electrons in the material are able to keep up with the wave and cancel it out internally. Real conductors let the waves penetrate a bit into the material because the electrons cannot perfectly respond to the incident fields. The measure of how much the fields penetrate is the "skin depth". If the object is much thicker than the skin depth at a certain frequency, then no waves get through to the other side. If the object is much thinner than the skin depth, then the wave can get through. The skin depth generally gets smaller for higher frequencies, so that lower frequencies can get through thicker objects.
A non-conductor will have a dielectric response, which means the bound electrons move in response to the EM wave and interact with it. The bound electron response is frequency dependent. Below a resonant frequency of the material, the waves pass through. Near a resonant frequency, the material's electrons respond strongly, and there can be strong refraction, absorption, and reflection of the wave. Above the resonant frequency, things quiet down again and the wave can go through. Real materials have many resonant frequencies whose effects all add up, so it gets complicated pretty quickly
At high enough frequencies, whether a conductor or not, the electrons cannot keep up with the wave and so there is very little interaction. The waves pass on through as if there was nothing there.
Say if it is copper, what frequency is high enough that it will get through? And at what thickness? can you give an example?
If the object is not a solid slab, but has holes, then you have the additional possibility of the waves going through the holes. The holes will act as waveguides. But waveguides only allow through waves about the cutoff frequency. Loosely speaking, long-wavelength waves cannot fit through small holes, so lower frequencies are reflected by an object with holes while higher frequencies are not.
What is the formula for frequency vs the radius of the hole?
The ability of a Faraday cage to reflect waves of a certain frequency depends on the thickness of its walls, and the size of its holes compared to the wavelength.