The problem
Combustion gases in a liquid rocket engine reach around 3,000 K or more, far above the melting point of any practical wall material. The wall survives only because heat is pulled out of it continuously, so that its hot side stays at a temperature the metal can tolerate.
Where the heat goes in
Heat transfer from the gas to the wall depends strongly on gas density and velocity. Bartz’s 1957 correlation, still the standard first estimate, shows it peaking around the throat, where the gas is still hot and moving at the speed of sound. Designers therefore make the cooling channels narrowest and the coolant fastest there.
How the channels are made
Traditionally, channels are milled into a copper-alloy liner and closed with an outer jacket. Metal 3D printing now allows channels to be built inside a single part, with shapes that change continuously along the engine. NASA has printed channel-cooled chambers from the copper alloy GRCop-42 and hot-fired them.
When it does not work well
Small engines are harder to cool regeneratively: the wall area is large compared with the propellant flow, so the coolant can absorb less heat per square centimetre of wall, and the channels become tiny. Small thrusters therefore often use other methods, such as a film of fuel along the wall, heat-resistant materials that radiate the heat away, or ablative liners.