Heat
Heat is thermal energy transferred from a warmer object to a cooler one by conduction, convection, or radiation, measured in joules.
Definition
Heat is thermal energy moving from something warm to something cooler, and it always flows from hot to cold, never the other way on its own. It is the transfer of thermal energy between objects or regions at different temperatures, measured in joules, and it moves by conduction (direct contact), convection (movement of fluids), and radiation (electromagnetic waves), continuing until both objects reach the same temperature, called thermal equilibrium. Formally, heat $Q$ is energy transferred across a system boundary because of a temperature difference, distinct from work $W$, which is energy transferred by macroscopic forces, and the first law of thermodynamics reads $\Delta U = Q - W$ with $W$ the work done by the system. Conductive heat flow follows Fourier's law, $\dot{Q} = kA\,\Delta T/L$, and radiative transfer follows the Stefan-Boltzmann law, $P = \sigma \epsilon A T^4$. The second law forbids spontaneous heat flow from cold to hot.
Example
When you hold a mug of hot chocolate, heat flows from the mug into your hands and warms them, and heat flows from your warm hand into an ice cube and melts it. A $0.5\ \text{kg}$ iron pan heated from $20^\circ\text{C}$ to $180^\circ\text{C}$, with specific heat $450\ \text{J/(kg}\cdot^\circ\text{C)}$, absorbs $Q = 0.5 \times 450 \times 160 = 36{,}000\ \text{J}$. A $1\ \text{m}^2$ window pane $5\ \text{mm}$ thick, with glass conductivity $k = 0.8\ \text{W/(m}\cdot\text{K)}$, separating $20^\circ\text{C}$ indoors from $0^\circ\text{C}$ outdoors passes $\dot{Q} = 0.8 \times 1 \times 20 / 0.005 = 3{,}200\ \text{W}$ through the glass alone. That is why still air layers and double panes matter so much.
Key Insight
Cold is not a thing that flows. When your hands feel cold on ice, it is heat leaving your hands, not coldness entering them. An object does not "contain" heat either; it contains thermal energy, and heat is the energy in transit, the same way "rain" is water on its way from cloud to ground. Heat and work are both energy in transit, and neither is a property of a system. A gas can gain the same internal energy by being heated or by being compressed, and once inside, the energy carries no memory of how it arrived.