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Thermal energy is the energy in a system which arises from internal motions. The "hotter" something is, the greater its thermal energy. As something absorbs heat, its thermal energy increases.
To feel something as hot or cold is actually feeling the transfer of heat from the object with a higher internal kinetic energy flowing to the object with a lower internal kinetic energy.
If you have ever gotten a carpet burn, you have experienced the conversion of kinetic energy into thermal energy via friction. As you slide across the carpet, your kinetic energy is converted into thermal energy, which is why your carpet burns always feel warm.