Why do processors often need fans and/or heatsinks, or why do devices sometimes get warm in areas?

The CPU's, System on Chips and Graphics Processing Units typically get hotter while they're processing heavier work loads or "clocking up" to work faster. Like other electronic components (except superconductors), some of the electrical energy flowing through them gets converted to thermal energy thus they always give off excess heat, even if miniscule. Since microprocessors have lots of tiny wiring, transistors and other components packed in such a small area, this heat gets concentrated. When more sections or areas of the chip gets used thus powered up, more heat has to be expected, while increased switching (due to the increased clock) leads to more changed/transitioning states (Switching Power) and a need to increase voltage to compensate for less time for signal propagation and gate transitions.

With great heat comes with great problems. Even a modest amount of excessive heat can change the electrical properties on electronic components, making the system unstable or unreliable. With enough heat, permanent physical damage will occur. Heatsinks (chunks of metal) and heat pipes (heatsinks with some kind of fluid inside) are used to draw this excess heat away from the processors, while fans help cool said heat conductors typically with air. Water cooling works similarly, although it uses the flow of a liquid within a pipe and chamber (or hollow heatsink) to transport the heat away to a more remote heatsink and fan array or other cooler.

Warm spots in devices like tablets and smartphones are typically caused by the SoC (or CPU) doing lots of processing for a long enough period. However in rare cases it can be caused by the battery, power controllers or other faulty components. Please bare in mind that an overheating lithium-ion battery is quite dangerous and it or its charger should be replaced or repaired by a professional ASAP - do not resume charging or using a device with any faulty batteries, power controllers or chargers.