Why is the OS design able to reduce power consumption?
Common CPUs that go into smartphones, laptops and even desktop PCs have a variable clock rate. When the scheduler detects that it has idle time, it can reduce the clock rate, and increase it again if there are more processes competing for CPU time.

CPUs optimized for battery-powered devices tend to be composed of many functional components that each have their own clock (example: ARM Cortex A8). When one component is idle, the operating system can turn it off (or reduce its clock), which saves power. For example, on many multicore CPUs (such as the ones in high-end mobile devices), each core can be turned on or off separately. ARM is particularly good at this game, which is one of the reasons why most smartphones have ARM CPUs.

Varying the clock rate or turning off the clock of electronic components is known as power management. It tends to be a complex part of writing and optimizing an operating system for a battery-powered device, with many dependencies on the exact hardware model.

What goes for the CPU goes for peripherals too. The backlit display is a major consumer of power on a smartphone, for example, followed by the radio. The CPU operating system may play a role in power management of peripherals; secondary chips also runs their own firmware which performs power management as well.

Applications have their part to play in keeping power consumption down: they must allow the operating system to do its thing. The worst thing an application can do is polling — running a loop like while (not_ready()) {}. Even introducing a small delay as in while (not_ready()) {usleep(100);} doesn't help as it doesn't allow enough time for the processor to go into low-power mode, or if it does, each unfruitful wake up means wasted energy. Thus the operating system APIs must be designed so that applications never need to poll, but can instead subscribe to some kind of event mechanism and remain idle until they are notified of a relevant event. Applications in turn need to take advantage of such mechanisms, so the design of the whole software stack has an impact on power consumption.

You can get some information about what's responsible for your PC's power consumption with Intel's Powertop utility. Smartphones typically have a way to see how much power applications have been consuming as well. Accounting for power consumption by application precisely is difficult: if a component wakes up for two applications, the wake-up time may be accounted to one of them somewhat arbitrarily or to neither; peripherals' power consumption are also not always be easily tracked to the application responsible.

A RAM chip doesn't know which of its bits store data of an active process, so it can't be turned off selectively in this manner. A process's power consumption isn't related to the amount of memory that it uses (except inasmuch as the RAM accesses consume power, but re-using the same memory or using different RAM areas makes no difference with respect to power consumption).