Conduction in Solids--way1

When one end of an object is supplied with thermal energy, the molecules gain kinetic energy and vibrate vigorously. As they vibrate faster, they will collide with neighbouring particles making them vibrate too. Due to the transfer of kinetic energy, the less energetic molecules would vibrate faster and collide with other less energetic molecules in the other end.
This continues until heat energy from the hottert part spreads throughout the colder part.

Energy is passed on from one molecule to another without the molecule moving out from their fixed position. Conduction also happens in all matter, as long as there is a medium.

As conduction transfers thermal energy by the vibration of particles, the closer the particles are packed togther, the more quickly it is for the energy to pass from one particle to another via vibration. That is the reason why conduction occurs the fastest in solids, as compared to liquids and gases.