Two words that get mixed up
Temperature is how hard the particles of a thing are moving. In a solid, that means how vigorously they vibrate about their fixed places. Hot metal has atoms rattling hard; cold metal has atoms barely stirring. A thermometer reports that motion as a number.
Heat is energy on the move from one thing to another because of a temperature difference. It always flows from hotter to colder, never the other way on its own. A cup of tea does not have “heat” sitting inside it; it has a temperature, and heat leaves it into the cooler air until the two match.
Passing it along
Look inside the bar. The particles next to the flame rattle hardest. They jostle their neighbours, which start rattling harder too, and those jostle theirs. Nothing travels down the bar except the vibration itself, handed from atom to atom. This is conduction, and it is how heat moves through solids.
Slide on to the heat map and the handover shows up as colour creeping from the hot end. The curve above the bar is its temperature at each point: steep near the heater, flattening as the far end catches up.
Try it: set the heater to Hot with the bar made of Copper, wait for the far end to warm, then choose Wood and watch the colour stall.
Why metals are cold to touch
Copper carries heat along at a speed that glass and wood cannot approach, because in a metal there are loose electrons free to carry energy as well as the vibrating atoms. That is why a metal handle burns you and a wooden one does not: the metal conducts the heat of the pan into your hand fast, the wood barely at all.
It is also why metal feels cold at room temperature. It is no colder than the wood beside it; it just pulls heat out of your fingers faster, and that flow is what you feel.
Two other ways heat travels
Conduction needs things to touch. Heat has two other routes.
In a liquid or gas, warm regions expand, become less dense and rise, while cooler regions sink to take their place. The circulating flow carries heat with it. This is convection, and it is what stirs a heated pan, drives the wind, and moves the warmth of a radiator around a room.
The third route needs nothing at all. Every warm object gives off invisible radiation, and the hotter it is the more it gives off and the shorter the waves. That is how the Sun’s heat crosses empty space to reach us, and why you feel a fire on your face from across a room.
Where the warmth goes
Heat always spreads out. Left alone, a hot bar in a cool room ends up at the room’s temperature, and the room ends up imperceptibly warmer. The energy is not lost, only shared out so evenly that it can no longer do anything useful. Every engine, every living body and every star is, in the end, a way of putting off that evening-out for a while.