A Mach diamond is the banded pattern that appears in a rocket or a jet engine's afterburner exhaust when the exhaust is moving supersonically.
It's essentially a visible sound wave, within the supersonic gas.
It is caused when the pressure of the exhaust is slightly too high (underexpanded) or too low (overexpanded).
Roughly speaking, what happens is that the supersonic gas leaves the nozzle and tries to expand/contract to the ambient pressure.
Because it is moving so fast it cannot mix with the surrounding air, so the gas jet changes diameter; however momentum causes it to overcorrect, and then the expansion/contraction reverses; it will typically repeat this expand-contract-expand cycle several times until it eventually damps out, in the meantime it forms the banded pattern that you can see.
The banded pattern is actually a shock wave that is associated with, but not exactly at the maximum pressure point.