A Japanese word, it means the quality of the out of focus part in photography.
It depends on: the shape of the diaphragm and on the optical design of the lens.

Apparently, the Japanese consider this a very important factor in lens choice.

Technically, it depends on the shape a point of light becomes when it is out of focus. Theoretically, it should turn into a flat disc of uniform brightness. In reality, the brightness varies. If you graph the brightness of the out-of-focus disc, you can have various profiles, with forms that resemble Mexican hats (desired) or donuts (undesired).
A donut profile indicates a lens that that turns unfocused points of light into quasi-rings (that's to say, discs with a circumference much brighter than the center). As a consequence, these lenses turn an unfocused bright line into two lines, which is considered hateful.

Bokeh is one of those things that you don't see until someone points them out to you: once you have seen it, it will never go away, and will make you unhappy about your cheap lenses for ever.

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