I have seen people mention (reference) "monitors" when talking about equipment for (semi)-professional musicians, but when I google the products I find that they look just like the boxes that we used to call "loudspeakers". What is the difference, if any, between the two?
There are (at least) two types of speakers referred to as monitors, and (at least) two types of speakers referred to as loudspeakers:
studio monitors -- speakers that are specifically designed to have flat response, minimal distortion and so on. Their intended use is in the context of a recording studio to provide unadulterated playback of the recorded material.
live monitors -- speakers that are used during live performance (sometimes for playback during recording) so that the performer can hear what they, and the rest of the ensemble are doing. These need to be loud enough for the situation at hand, and need balanced output, but don't need the pristine characteristics of studio monitors. Often, these are wedges positioned at the front of the stage, projecting back towards the ensemble.
live loudspeakers -- the speakers that are used to project the sound of the performer out to the audience. More often these are referred to as "PA". Here overall power, and distortion free operation are the prime considerations. Some deviations from perfectly flat frequency response are sometimes acceptable, esp. if other speakers, e.g. sub-woofers, can fill in the missing signal.
stereo loudspeakers -- the speakers used to project the sound for home audio systems. These are tuned to produced sound that is more pleasing for listening entertainment.
The important difference between stereo loudspeakers and studio monitors is that the former are often voiced in a way that affects sound away from what is recorded; e.g. some may be bass heavy, or have a smile shaped frequency response curve etc. Ideally, studio monitors do not introduce any color of their own, but, again ideally, perfectly reproduce the recorded sound being played back on them. This reflects the different purposes: the former are designed for enjoyment, the later are designed for critical listening.
Good, clear answer by @Tetsujin.
Just one thing to add: when musicians talk about monitors in a live setting, this can refer informally to foldback monitors, also known as stage monitors. These are rear facing loudspeakers (usually) which allow musicians to hear what they are playing, and a mix of the other musicians playing with them. This allows musicians to have a clear mix of whichever instrumental/vocal parts they want to hear, and also means that musicians don't have to rely upon listening to the main PA speakers, which are usually in front of them anyway.
It's really just a matter of degree.
Monitors are loudspeakers, but you would expect them to be flat, clean & accurate, just the thing to use in your studio - & consequently expensive; whereas generic 'loudspeakers' could be the things in your boom box, or even your alarm clock, right up to your hi-fi.
Often they don't look particularly impressive compared to expensive hi-fi, because they aren't designed to look good, but sound accurate.
As mentioned in other answers there are different uses of the word "monitor". The most common usage in music production is the special speakers the sound engineer / producer uses to listen to the mix.
In technical terms what separates these from normal speakers is a flat frequency response.
What does that mean?
If you look at the fourier transform (the frequency domain) of a balanced frequency spectrum (the full range of frequencies from low to high at the same amplitude (loudness)) you will see spike spikes at each frequency present. If you play that signal through a pair of small cheap speakers - recording the signal as it comes out of the speakers and then again look at the frequency domain you will most likely see that the spikes at the high frequencies are a bit distorted (bigger or smaller) from the org signal but the spikes for frequencies at the bass end (low freq) will have much less amplitude - indicating that the speakers have unintentionally removed bass frequencies (small speakers can't reproduce all the long waveforms in bass frequencies). Other speaks might distort (boost or cut) the higher frequencies.
Expensive studio monitors however should produce a freqency domain spectrum very similar to that of the original signal - the signal has not had any frequencies boosted or cut by the monitors - they faithfully reproduce the original sound by maintaining the same power at each frequency present in the signal.
As a volunteer sound guy, I feel like I have to weigh in here. I think Tetsujin and Dave have explained the technical side very well, but there's also a practical side:
If you want good sound, you must keep the stage as quiet as possible!
Even for rock concerts.
Most performers will ask for more of something in their (foldback) monitors because they can't hear it. Maybe themselves, maybe someone else that they need to sync with, etc. If the sound guy only turns things up like the performer asks, there will be a point where the PA is completely worthless because the monitors are actually that loud. It can also be difficult to mix the house when the mics have so much "background noise" from the monitors, not to mention that annoying and embarrassing feedback squeal.
Instead, a good sound guy will also listen to the monitors himself, to determine what the performers might be hearing too much of, and turn that down.
In-ear monitors are nice too. They keep the stage dead quiet except for the performers' own, direct sound. But the sound guy should still listen to them, one at a time, through the console's headphones, so that he can turn things down as before and avoid breaking the performers' ears.
Monitor speakers are speakers that play the plain sound without distortion through the frequency range. They are designed to allow you to hear exactly the electronic sound.
"Monitor" is a specialist type of loudspeaker.
Most loudspeakers do not provide such a uniform frequency response.
Most loudspeakers have all different kinds of frequency response, but it is not uniform.
So the only special thing about a 'monitor' loudspeaker is that it has a flat and uniform frequency response throughout the range.
As an aside, this flat frequency response comes usually at the expense of efficiency. Take for example a guitar cabinet, it will be very loud for the power in watts that goes in, this is great efficiency of power in -> sound out.
But to make a monitor loudspeaker it typically requires a great sacrifice in volume; monitors need more power in to deliver the same volume of perceived loudness. They tend to need to sacrifice efficiency in the design, in order to be capable of achieving the 'flat frequency response'.
So a speaker with colour isn't a studio monitor but if a tone deaf person were to describe a loudspeaker of significant colour as being flat in frequency response and it were an expensive speaker then it could be determined to be an expensive studio monitor that is flat. My summary is akin to wine tasting professionals who when given blindfolded taste tests these proclaimed wine tasting pros cannot distinguish between an aged Italian or French vintage or a bottle of three buck chuck from your local grape grower. So taste, like sound, is in the pallet or oto of the beholder and how they can spin a raconteurs description of their experience.