Single Sideband is basically AM 2.0, so to talk about it in detail, we have to take a closer look at good old fashioned 19th century AM.
The graphic above is from Wikipedia. The top graph, the "baseband" signal, is the audio, aka the signal coming out of the microphone. It's vaguely what a human voice would look like on a specrograph.
The second graph is what AM looks like. The spike in the middle is called the "carrier." Let's say you're transmitting on 5 MHz with a 10 watt radio. When you push the talk button and then say nothing into the mic, you start broadcasting a 5 MHz wave with a power of 10 watts. A receiver tuned to 5 MHz will hear the background static go away, because you are transmitting a carrier wave that does not modulate much louder than the background noise, you are effectively holding the receiver's speaker still. A signal coming in strong enough to do that we call "full quieting."
The carrier carries no information. Another way to look at these graphs is, picture the sine wave. You may have seen something like this before:
That middle waveform represents AM, notice how there's always some squiggles going through the middle, and it varies toward the top and bottom edges? That's what the upper graphic is representing, that big center spike is always there for that reason.
You may notice that the two lobes to either side are the same shape as the base band signal; or one of them is, the other is a mirror image. We call these sidebands. That's actually where the audio is. In the second graphic, you can see how the top and bottom edges of the AM waveform resemble the baseband signal. Turns out, AM radio uses twice as much bandwidth and more than twice as much power to transmit the usable signal.
So what if we built the transmitter to only transmit one of the two sidebands and suppress the carrier and the other sideband? The same audio information goes over the air, and we take up less room on the radio spectrum to do it. What's more, since we're transmitting less overall "stuff," the radio's power is more focused on the part we do transmit, so a single sideband transmission comes across as "louder" than an equivalent AM transmission.
There are some cool upshots to how SSB works: the first is that the radio uses less power and overall stays cooler. AM (and FM) transmit with their full power all of the time, doesn't matter if you whisper or scream into the microphone you're putting whatever power your amplifier is set to out to the feed line. You might be transmitting silence, but it is very loud silence. SSB doesn't do that; the louder you talk into the mic, the more power goes out the antenna. You aren't constantly transmitting that carrier, so if your mic goes quiet, so does your antenna. Thus, your transmitter gear takes less power, runs cooler, and if you are on some consumable power source like batteries, you can transmit more effective power for longer.
Even cooler than that is the lack of collisions. If you've played with radios much, even listening to music radio stations near the edge of their ranges where you can kind of hear both, you know they interfere with each other. Happens all the time with aviation radios, pilots will transmit on the same frequency at the same time and anyone else listening gets to listen to the psychedelic sounds of two carrier waves interfering with each other. On FM radios usually the louder signal "wins" and the other one just sounds like static or interference under it. SSB doesn't do this.
If two people transmit at the same time on the same frequency on SSB, a third person listening just hears two voices, just like if two people talked at the same time in a room. Hams hold contests to see who can make contact with the most people from the most places, and folks from somewhere rare will end up asking for contacts, everyone else says their callsigns at once like it's the floor of the stock exchange, the one station will pick someone he heard to exchange details with, rinse and repeat. It's 1950's Discord.
Hams also use this technique to send text back and forth extremely efficiently. If you tune your software defined radio to 14.070 MHz (or use one of several people make available online) you might hear what sounds like several strange warbling whistles that come and go. That is PSK31, a digital text mode designed to use an ordinary PC sound card as a modem, and an ordinary SSB transceiver to send the signals across the air. Using software like FLDigi, you can receive and transmit text over the air, and each text transmission is very narrow in bandwidth. Over a dozen can take place in the same space as a normal voice channel, you leave the radio tuned to 14.070 and choose which transmission to listen to by clicking on them in the waterfall, ie choosing what audio frequency to listen to, which only works using SSB because no carrier collisions.
For all its advantages, there are some disadvantages. You cannot transmit "quiet" with SSB the way you can with AM or FM; so the background static, the "noise floor" is always there. Makes it not so nice for listening to music, which is why you basically only ever see it on communication radios. And it also requires a much more complicated transmitter and receiver while achieving the same or slightly worse audio quality than AM. You don't see SSB used much in VHF and above because there's so much room for activities/line of sight limits how far your signal goes so the efficiency advantages of SSB are less important, which is why we tend to use FM (or AM for old shit like airplanes) at wavelengths shorter than 6 meters or so.
