In the 1960s, many PA systems were finally of high enough quality that bands had excess speakers, and naturally, began turning them around to hear themselves better. These primitive monitoring solutions were the start of an entire industry: an industry devoted to helping musicians hear better. In recent years, in-ear monitoring - the use of tiny speakers in your ears - has been gaining popularity over traditional monitor speakers both on stage and in home studios. Both offer their strengths and weaknesses.
Wedge - or speaker - monitoring is the standard for both home studios and live music clubs. In live sound, the wedges are fed from either a separate monitor board, which takes a split from the stage and creates a custom mix for each musician, or fed from the auxiliary sends of the front-of-house sound board. Monitor wedges tend to be very loud; they're credited with being one of the reasons working musicians have to be so conscientious about their hearing health. The advantages to wedges are pretty clear cut — a lot of musicians prefer wedges because it allows them to form a custom listening environment that includes not only the wedge and the mix coming from it, but their guitar amps and the reverb of the room. However, most audiologists agree: the loudness of wedges is bad for your hearing. Wedge monitoring is also difficult for working bands who have to provide their own PA systems; the systems are heavy and require a lot of setup time.
In recording studios, speaker monitoring is the standard. It's challenging to get a good mix on headphones unless they're very flat and accurate. Speaker monitoring is also the best way to see what a mix will sound like on a variety of systems.
In the early days of in-ear monitoring, engineers like Marty Garcia at Future Sonics were putting stock Sony earbuds in crude earmolds connected to hard-wired amplifiers. Now, 20 years later, we have extremely complicated in-ear systems; custom-molded earpieces with two or three speakers in each (to handle the mids, highs, and lows seperately) are becoming the standard, and many in-ear monitors are incorporating ambient systems into their earpieces to reduce the learning curve on in-ears. In-ear monitoring has several advantages, the greatest being hearing conservation. Cutting yourself off from loud stage wedges is a great idea, as you can control your volume and mix as you want it without a struggle.
The disadvantages are, surprisingly, similar to wedge monitoring: sometimes listeners push the in-ears louder than they should, forgetting that by doing so they can hit the same sound pressure levels as wedge monitors. In addition, a lot of artists can't get used to the isolation, which can be combated by using ambient microphones on stage.
For home studios, a good pair of in-ears can be an accurate - although expensive - way to monitor your recording mix.