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imagine a container with no gases inside. let's introduce some liquid here - it will evaporate until some kind of equilibrium is reached. pressure of vapor achieved is called vapor pressure and is strongly dependent on temperature. if pressure of gas above liquid is equal to vapor pressure, liquid is boiling.
you can make any liquid boil by decreasing pressure or increasing temperature, and conversely you can any gas condense by decreasing temperature or increasing pressure. if you look at a phase diagram there are regions where liquid or gas is stable. if you have liquid in "gas stable" area, it will evaporate, and if it's the other way around, it will condense.
let's cool down top surface of that container. vapor will condense on it, lowering pressure, making liquid on bottom boil, which decreases its temperature until it can't boil at a given pressure. more vapor condenses and eventually bottom part reaches temperature of our cold source. this device is called thermosiphon. this way we transferred heat from bottom to top of container. if you make it so that liquid covers entire surface of container, for example by use of capillary forces, this device can transfer heat from any orientation to any other. this is called a heat pipe.
if you want to transfer heat from colder to hotter area, you need to make liquid boil at lower pressure and condense at higher pressure. you need to put some energy into that and this device is called a heat pump. notice that the lower the temperature difference is, the lower pressure ratio becomes and the more efficient entire thing is. the trick is, at low temperature differences amount of energy needed to run the pump is small compared to amount of heat moved around. there's hard physical limit on this - it's inverse of maximum (carnot) heat engine efficiency at these conditions, it can easily reach 3x-4x in practical implementation.
you can even reverse this and extract useful energy from temperature difference. this is called heat engine and about all (non-hydro, non-wind and non-photovoltaic) electricity generation runs on it. you can even treat these renewable energy sources as a heat engines, in some cosmic sense - taking heat source and heat sink as temperature of sun and temperature of photovoltaic panel, for example (with additional limitations)