Power Meter
Power meters measure your power output. Power can be used to pace a ride, and to do structured interval training. This StackExchange question has more in depth discussion. Power meters are ubiquitous in professional road racing. They can benefit many amateurs, but consider the cost-benefit.
Alternatives include training by heart rate or by perceived effort. That said, heart rate will take several seconds to respond when you start an interval, and it will likewise take several seconds to come down when you stop an interval. Heart rate can be affected by fatigue. Some of the same criticisms hold for perceived effort. Power overcomes those disadvantages. That said, amateur riders do not need a power meter to progress satisfactorily.
For structured training, riders estimate their functional threshold power (FTP). There are several protocols to do this. This is the maximum power generated when your body is relying mainly on aerobic (as opposed to anaerobic) processes to generate energy. Fresh riders can sustain their FTP for about a 35 to 70 minute maximum effort. You can base intervals off their FTP, e.g. warm up, ride 2x 15 minutes at 94-100% of FTP with a 7 minute rest, then cool down. More information on training zones (by FTP and heart rate) is here. Riders need to re-estimate their FTP during their season to keep their power targets in the correct zone.
A brief history and basic technical information
Power meters use strain gauges, which measure the amount of deflection in a bike component, as well as a cadence sensor. This answer has more detailed information on how power meters work physically. They transmit this information wirelessly to a computer, typically over Bluetooth or a sport-specific protocol called ANT+. The data files can be analyzed by a coach and uploaded to a training site like Strava. Note that power meters require a computer capable of communicating via ANT+ or Bluetooth to work, which is an additional expense. The PowerTap shown below had a proprietary computer, but all current power meters work with most bike computers that have ANT+ or Bluetooth capability.
The first power meters were made by SRM (Schoberer Rad Metrik, a German company), and they were very expensive. As with many electronic goods, they have declined substantially in price in recent years. SRM power meters replaced the chainring spider, and several other power meters are still mounted here. Later power meters were mounted in hubs (e.g. PowerTap, now owned by SRAM, pictured below), the crankarms, or the pedal spindles, or a few less common arrangements. E-bikes also measure torque, a component of power, to determine how much assistance to provide.

It is preferable to measure the total power generated by both legs, but some manufacturers offer left-only power meters to save costs. Power meters mounted at the crank spider, the hub, or on both crankarms or pedals will measure a rider's total power. Left-only power meters can be placed on the left crankarm or pedal spindle, or even on the right side. Left-only power is less accurate than total power. Also, if you have two independent power sensors on each side, you can measure the power balance between left and right, but it isn't clear if this is a significant benefit versus just knowing total power.
Power also enables virtual cycling environments like Zwift, where you simulate a rider's progress through a course and interactions between multiple riders. Virtual cycling programs can accept readings from a power meter. A traditional trainer can be used to estimate power as well, although this is less accurate than an actual power meter. Last, smart trainers measure power at the trainer's flywheel, and they can vary the flywheel resistance to simulate changes in gradient on the course.
Maintenance
Power meters are sealed units. They don’t suffer wear in the same way that chains, cables, and other bike components do. If replaceable, the batteries will need to be replaced periodically. The strain gauges are delicate, and users should take as much care as possible not to knock the power meter unnecessarily.
Power meters may need to be periodically zeroed, I.e. have the software confirm the strain gauge reading when unloaded, like you would tare a weighing scale. Many newer ones can auto zero. If you suspect a power meter is still reading wrongly, another step may be to calibrate the slope, I.e. the change in resistance with a known weight. This is usually done at the manufacturer, although some power meters enable users to do this. Smart trainers have a spindown function that is equivalent.
Smart trainers contain drive belts and cartridge bearings. It is possible that these will wear out before the electronics fail. Both belts and bearings tend to come in standard sizes. You could take the worn belt in to an auto parts store. For bearings, you can measure the physical size of the bearing; see the entry on bearings for more information.
Reference:
Wikipedia