Thermal conduction is a very important mechanism for moving energy around. We insulate our attics and wear coats and hats as ways to prevent conduction; we buy aluminum and copper-bottomed pans because they conduct heat well; we use wooden spoons for cooking because they do not conduct heat well.
Thermal conduction explains why our bare feet make can feel either cold or warm when stepping on two different flooring materials both at room temperature. Tile feels cold because it is a thermal conductor and so draws energy rapidly away from our feet. A plush rug, on the other hand, is a poor conductor (a good insulator). It draws energy from our feet only very slowly, and so does not feel cold at all.
Here are some more examples of the varied contexts in which thermal conduction is important:
Why dew forms on the grass, and not on the sidewalk
What we can learn from snow patterns on roofs
What we can learn from the snow patterns on the ground
The unit on thermal conduction