Inductors are one of the three basic types of passive electronic components (along with resistors and capacitors). The purpose of inductors is to store electrical energy by creating magnetic fields and then reinducing the magnetic fields to electricity again, using coils. This is useful in several applications, especially power handling.
Boost and buck converters (which step DC voltage up and down respectively) are centered around inductors and some sort of switching device, some type of transistor in most cases. In the example of a boost converter, the goal is to increase the supply voltage. This is basically done by switching power through an inductor on and off very quickly, usually in the several hundred kilohertz range. When the power is switched on, a magnetic field is created in the same way as is used in electromagnets, solenoids, and electric motors. Very quickly, the power is switched off and that magnetic field collapses, inducing a current back through the wire even though the power has been turned off. The power is switched back on before the surge has completely died down, and these pulses of stored energy are used to charge up a capacitor to a greater voltage than the supply with a diode to prevent backflow. Note that the when the magnetic field "collapses", it doesn't do so all at once. There is an exponential decay.
When two inductors are coupled together in such a way that their magnetic flux fields line up, they form a transformer. A transformer isolates one side from the other, and can increase or decrease an AC voltage. A common use for isolation is in dynamic microphones, where a small transformer is used to prevent damage to the microphone if phantom power is applied, invert one signal for balanced output, and reduce annoying hum by "lifting" the ground of the microphone side.
More importantly, transformers are used in electrical power distribution grids to step the electricity generated at a power station up to many thousands of volts for more efficient long distance power lines and interties. Then, closer to homes and buisnesses the high voltage is stepped down by transformers to more useful voltages, usually 120 and 240 volts AC. If we had 250,000 volt outlets, it wouldn't be very fun to get zapped whenever we came within a few feet of one! Not to mention very expensive insulation on wires and sockets...
Most of us are probably familier with inductive, or "wireless" charging. This is a specialized kind of transformer formed by two inductive charging coils, a transmitter and a receiver. The difference is that they are not magnetically coupled like a typical transformer would be, and instead have a gap between them filled with glass or a bit of plastic or wood in the outside of each device. This means the efficiency is usually not as great as a conventional transformer, but through the use of advanced sensing, the transmitter of many charging mats is able to vary its output power and possibly its frequency and cater to the placement of the receiving coil and the distance between them.
Inductive charging is used in many cases, such as in electric toothbrushes that need to be waterproof, or phones which are trying to be more waterproof and portless. As one can imagine, there are plenty of advantages such as ease of use, fewer penetrations in body design, and less connectors to fail.
Nevertheless, there are disadvantages to wireless charging. Lower efficiency is one, and although that is constantly being improved, it still poses a difficulty when one or both sides are using limited battery power. With lower efficency, that wasted electrical power is converted mainly into heat, so care must be taken to protect against overheating. Also, placement of the receiving coil on the transmitting coil affects efficiency. Some products address this issue by using magnets or a keyed design to align the two. Lastly, space in the product being charged must be considered. A receiving coil doesn't neccessarily need to be that thick, but if it is going to provide any significant power transfer, it must be as large as possible with many windings. This is a factor when phone designers are striving for a thin and light body.
Another function of inductors is in antenna design dealing with radio frequency (RF). Chokes are commonly used to cancel out capacitance in an antenna which would be problematic otherwise. Circled in blue are several small resin-encapsulated chokes on the antenna side of this amateur radio transceiver I built. RF circuits can be pretty complicated but the basic reason for these chokes is to match resonant frequencies with the antenna which leads to the most efficient transmission.