If a metal spoon is put in a cup of hot tea, after some time its other end also becomes warm.
Heat is transferred from one end of the spoon to the other end due to molecular vibrations within the spoon. This process is called Thermal Conduction.
In this process there is no movement of mass.
In the beginning of conduction process, a part of heat is absorbed at every cross-section and the remaining part is transported to the next section.
But after time a steady-state is reached and the temperature of all the parts of the conductor becomes constant.