Single sideband (SSB) is AM with the carrier and one of the two sidebands thrown away. What remains is the information, in about half the bandwidth, with all the transmitter's power spent on it. It is the standard HF voice mode.
Why drop the carrier and a sideband
Both sidebands of an AM signal carry the same information, and the carrier carries none.
In AM the carrier is steady and tells the receiver nothing. The two sidebands are mirror copies of one another. Remove all that redundancy and you have a signal half as wide (so the receiver lets in half as much noise) and with every watt carrying speech.
A bonus: with nobody talking, an SSB transmitter sends nothing. Power follows your voice, so it is rated in PEP, the power at the peaks of the audio envelope.
Making and receiving it
Balanced modulator: double sideband, no carrier. Filter: one sideband. Product detector: back to audio.
A balanced modulator multiplies audio by the carrier and cancels the carrier, leaving two sidebands. A sharp filter then removes one. (The phasing method does the same with maths and a 90° shift, common in SDR.)
The receiver must put the missing carrier back with a local oscillator in a product detector. If that oscillator is off by a few hertz, every voice tone moves by the same few hertz, and the voice turns thin or "Donald Duck". You fix this by tuning, or with RIT.
Upper or lower
Choose one sideband to send. The carrier and the other side are left out.
The audio is placed above the carrier frequency (USB) or mirrored below it (LSB). Which one you use is a pure convention: LSB on 160, 80 and 40 m, USB on 20 m and up, and USB on 60 m, so that both ends of a contact agree. The dial shows the suppressed carrier, so the signal extends 3 kHz away from the number on the display. On a band edge, keep the whole signal inside.
Trade-offs
Needs accurate tuning and a stable oscillator.
Needs linear amplifiers, because distortion makes intermodulation splatter. See Linear amplifiers.
Overdriving the microphone gain does just that: flat-topping, wide, ugly signal.