Simply put, ambisonics is a way of representing sound in full three dimensions. It captures audio as one omnidirectional signal and three figure of eight signals, with the microphones facing left/right, front/back and up/down.
It was a process invented in the 1970s, but has really become practical with low cost, simple digital systems like the Brahma microphone. It is more complex than using a single-point stereo microphone, but the results are very much more rewarding.
In conventional recordings, much of the quality of the recording is determined by what microphones you choose (Omni, cardioid, hypercardioid) and how and where you place them. This gets frozen once you record.
In an ambisonic recording, much of this is still in your control after the recording. You can create virtual microphones of different patterns, angle and rotate them.
You can set up with eight or more loudspeakers and accurately recreate the sound you heard at the performance. Set up four and hear it without the height element. If you already have a 5.1 system, you can synthesise a 5.1 recording that is actually better than you would get with a multimicrophone system.