Things to consider when choosing gasket materials:
- Type of fluid/gas that should be contained
- Temperature range
- Roughness of the sealing surfaces. The gasket needs to fill impurities in the sealing surfaces to keep the seal from leaking
- Mechanical friction (Both for efficiency and resistance)
Perhaps the last one concerns other kind of seals more, on axles for example.
All these combined gives that a gasket needs to be chosen as a compromise. On a rough surface, you might want a thicker, softer gasket, but that won't work on high pressure applications. It might need to resist oil or fuel or keep drinking water uncontaminated.
Valve head and oil pan gaskets are often thicker and softer. I've seen cork and rubber for example, and they won't need much torque to seal. You can compare this to the head gasket, which needs to hold the pressure from combustion and withstand a lot of heat. It often have steel reinforcements to accomplish this.
The head gasket needs a lot of torque to not leak, even though the surfaces are generally very smooth. The sometimes ridiculously specific torque specification are there to get the exact and even pressure on the gasket. Too little and it will leak, to much and something will be damaged (and probably leak).
You mention copper washers. A softer metal can be used between two harder metals to form a gasket since it will deform and follow the shape of the surrounding surfaces. (It's usually recommended to change the washer after every disassemble). In your case it was used on a Banjo connection. Imagine using a fiber gasket there, it would tear up when adding tension to the screw.
Many different materials are used for gaskets. Copper, Aluminum, plastic, rubber, cork and fiber being very common, but there are a lot of other types as well.