Linear power supply
A linear power supply is one that uses purely analog techniques (i.e. internal signals with voltages with continuously varying values) to convert input power to output, at the voltage needed. (The internal devices operating in this mode are said to be operating in their 'linear' mode.)
Linear supplies are generally inefficient because of the VI conversion issue; since linear supplies are usually quite simple internally, they usually do not have any mechanism for this. Generally, the extra power (the actual amount is given by the amperage multiplied by the voltage drop from the input to the output) is simply discarded, as heat.
The usual internal circuitry is that a combination of a transistor and the load (output) are used as a voltage divider network; the emitter of the transistor is connected to the output, and the collector to the input voltage. The current through the transistor's base is adjusted so that the voltage drop across the load is what is desired; the rest will be dissipated in the transistor.
Depending on the circuit details, in linear supplies the output voltage is often dependent on the input voltage (and directly related to it), so if the latter varies, so will the former. Simple linear supplies are thus unsuited to applications where precise regulation of the output voltage is needed.
Due to their low efficiency, in high-power applications they have generally been replaced by switching power supplies.