Which one of two materials will feel colder: one having high thermal conductivity and low heat capacity and other having low thermal conductivity and high heat capacity? Please see question details.

So to the best of my knowledge the feeling of "colder" is purely driven by heat transfer rate (ignoring physiological feelings of cold like menthol activated http://en.wikipedia.org/wiki/TRPM8 receptors) or the feeling of heat being sucked away from your body.  For an instantaneous touch, neither conductivity or heat capacity should really play a role. Heat transfer rates can be classically defined by q = h dT/dx where h is the heat transfer coefficient and dT/dx is the temperature gradient.  So given your particular situation, the gradients are equal.  The heat transfer coefficient, which can be congruent with the thermal conductivity but is not necessarily so, is the main driver. However to answer your question which is, presumably about sensation over time, start by referring to thermal http://en.wikipedia.org/wiki/P%C3%A9clet_number and the http://en.wikipedia.org/wiki/Nusselt_number.  They should help you figure out the magnitude of the various parameters, which is what your questions is focused on.  This should tell you which property is important. Also other factors should affect sensation of cold, i.e. velocity in convective heat transfer, latent heat of fusion/vaporization/sublimation etc associated with phase changes.

The material with high conductivity will feel colder. For simplicity sake, I will call the material with high conductivity "metal" and the material with low conductivity "plastic". Two concepts are useful to understand why: The only thing that specific heat capacity does is slow the response time until the system reaches steady state; it does not in any way influence the temperature distribution at steady state. Its only influence will be in a transient state Thermal conductivity alone will dictate the temperature distribution at steady state Human skin does not feel temperature, it feels heat flow and heat transfer, like heat gain or heat loss So, when your finger touches both materials: at the first moment, the metal will conduct much more heat within itself than the plastic, because it is a better conductor once it reaches steady state, the metal will still be colder because whatever effect the specific heat capacity had during transience, it no longer has any influence the only time where it is possible that the plastic would feel warmer is the time between first contact and when it reaches steady state, where it will depend on many things

Thermal feeling is instantaneous, for evolutionary reason so before the heat is dissipated in a material, the (higher) thermal conductivity of that material already causes the thermal sensors in the skin to register a (bigger) drop in thermal energy and the brain interpret that as a cold(er) feeling.

the sensation of an object being cold or hot depends upon both thermal conductivity as well as specific heat, which can be represented by thermal diffusivity. it all depends on for how much time u are in contact with the object. Here thermal conductivity rides the extent to which u feel cold or hot and specific heat defines for how long will u be able to feel that sensation, surely it degrades with time. for example water has a high specific heat so for the same temp difference as with other objects of lesser heat value, u will sense heat interaction with water for a longer time. but when we are in contact with the object for a short period of time thermal conductivity is the only factor which will affect the sensation.

Alpha= K/density*specific heat The term alpha designates how fast a system can diffuses through material. The lower term in equation represents specific heat , how mush energy can it store.hence a material with high thermal conductivity but low heat capaity will have large thermal diffusivity.Hence from these type of material we get more cold effect.

Let us take example of two substances : Copper ( high thermal conductivity and low heat capacity) and Water ( low thermal conductivity and high heat capacity) When in contact copper will give a much colder effect as compared to water because of higher thermal conductivity. On the other hand, water will give the cold effect for longer duration since it will take time to achieve equilibrium with surrounding because of lower thermal conductivity and high heat capacity.

For this we have a variable known as thermal diffusivity which is a ratio of (thermal conductivity/(density*specific heat)). More the thermal conductivity more easily heat can be transmitted and lesser the specific heat, less heat is absorbed by material i.e.; more heat is available for further conduction... So material with high conductivity and low specific heat seems to be colder since heat get quickly diffused instead of accumulation at a place like for material with low conductivity and high specific heat... So more the diffusivity material will be cooler..