Conduction
Conduction is heat transfer through direct contact, as faster-moving particles collide with slower neighbors and pass energy along.
Definition
Conduction is heat moving through something by direct contact: fast-moving particles bump into slower neighbors and pass energy along without the material itself flowing, one neighbor to the next, like a line of people passing buckets. It transfers thermal energy through a material, or between materials touching each other, working best in solids, especially metals, where free electrons carry energy quickly, and poorly in gases. Steady-state conduction obeys Fourier's law, $\dot{Q} = kA\,\Delta T/L$, where $k$ is thermal conductivity, $A$ is cross-sectional area, $L$ is thickness, and $\Delta T$ is the temperature difference; in insulators and gases, energy instead travels through lattice vibrations (phonons) or molecular collisions, which is far slower.
Example
Leave a metal spoon in a pot of hot soup and the handle gets hot as heat travels up it by conduction; a metal chair in a $20^\circ\text{C}$ room feels colder than a wooden one because it conducts heat away from your skin about $1{,}000$ times faster, even though both are at $20^\circ\text{C}$. A copper rod ($k = 400\ \text{W/(m}\cdot\text{K)}$) $0.5\ \text{m}$ long with cross-section $2 \times 10^{-4}\ \text{m}^2$ connecting a $100^\circ\text{C}$ bath to a $0^\circ\text{C}$ bath carries $\dot{Q} = 400 \times 2 \times 10^{-4} \times 100 / 0.5 = 16\ \text{W}$, while a glass rod of the same size would carry only $0.032\ \text{W}$.
Key Insight
Metals conduct heat well for the same reason they conduct electricity: their free electrons carry energy quickly through the solid, and the same electrons explain the Wiedemann-Franz law linking thermal and electrical conductivity. Thermal resistance $R = L/(kA)$ adds in series exactly like electrical resistance, and that analogy is how building engineers compute the R-value of walls made from multiple layers.