Exactly what the name says, a device or equipment to measure "phasors", and something more: PMUs usually can measure the phasors at the fundamental and harmonics, can timestamp accurately the acquired data, can interface with a range of industrial buses and communication standards (e.g. CANbus, or Modbus, Ethernte, and similar), can trigger alarms when some kind of threshold is overcome, and can record also signals in time domain.

The topic is broad, many papers are/were written on the topic, regarding technical implementation (I wrote mine as well!), standards (look at IEEE std. C37.118 for example), applications, and so on. Depending on your field of study, needs, applications some aspects may be more relevant than others: e.g. accuracy/uncertainty.

Regarding it you have the acquisition itself, where accuracy regards digitizing hardware (e.g. linearity. number of bits, etc), but also stability of clocks and references. If you go to the algorithms for post-processing any things were said on the use of FFT (and DFT, CZT, etc etc), frequency leakage, interpolation, frequency stability: when you answer "the amplitude of 5th harmonic is 1.709 Arms" your figure is deemed by several sources of error (and uncertainty), depending on the sampling frequency, if the 5th is really the 5th (that is if mains frequency is stable and you get it accurately*), if leakage from nearby components is polluting your estimate, if the phenomena are stable or you have time varying harmonics .... there are standards such as EN 61000-4-15 to estimate these things following a procedure (not the best one, but it is a procedure).

* for railway systems the stability of the fundamental is an issue and has impact when you evaluate the harmonics; I wrote a few papers with my friend Rado Lapuh and David Slepicka on its estimate and the impact on spectrum estimates.

And now probably the best feature of PMUs and distributed measurement systems in general: synchronization of remote units or in other words absolute timestamp. If you want to study accurately an electric network you need phase differences between phasors that are reliable; so I am not talking that the reading taken at 12:00 was really 12 o'clock and there was no error with the wrist watch or daylight saving time issues :-), but accurate timestamping down to microseconds (in some cases even better). Why? think of the phase difference between two sinusoidal components at the same frequency (e.g. 2000 Hz) in two remote places at the two ends of a transmision link and you want to study stability: you tell yourself that 1° error in phase readings is a good starting point, that at 2000 Hz is 1.39 us! So, going to small errors and frequencies higher than fundamental is quite demanding. Methods: once we decide that hooking at the mains fundamental and its stability is not enough (by the way we want to study e.g. its instability!), there are a few possibilities, network timestamp (supplied by internet, subject to a maximum resolution around 1 ms, and a lot of jitter with respect  to when the 1 ms "flag" pops up in the software at the high application level, rather than passing through your ethernet card); so we need something low level, and there jitter and delay is even more troublesome (the IEEE 1588 protocol requires dedicated hardware to capture timestamp signals at the lowest level possible, reducing jitter and uncertainty, reducing the number of routines and hw/sw layers before you get your synchronizing signal); but, what is giving us the "real effective time"? I can synchronize several units with a network, a wire pair, a wireless signal (with different issues of transport delays, randomness, etc.), but I need one true time: this is the GPS and one station (the master) with a GPS, distributing accurately it to the other stations (so to other PMUs).

It's not that simple, but this is roughly the overall picture. Apologies if I have missed some important elements.

Phasor Measurement Units

Phasor Measurement units - called Synchrophasors - are precise grid measurements of electrical waves to determine the health of the electricity distribution system. 

Phasor measurement units are considered one of the most important measuring devices in the future of power systems. A PMU can be a dedicated device, or the PMU function can be incorporated into a protective relay or other device.

PMU measurements are often taken at 30 observations per second (compared to one every 4 seconds using conventional technology). Time stamping each measurement to a common time reference (provided by very high precision clocks) allows synchrophasors from different locations and utilities to be synchronized. When data from multiple PMUs are combined together, the information provides a precise and comprehensive view of the entire interconnection. Synchrophasors enable a superior indication of grid stress, and are often used used to trigger corrective actions to maintain reliability.