Octane is a measure of the ability of gasoline to withstand compression without spontaneously igniting.
Back in the old days, preignition in gasoline-engines was a major problem. This phenomenon, known as knocking, affects performance, but has largely been eliminated in modern gasoline engines. Today, high octane is mostly desired by high-performance engines, such as gasoline turbo engines.

Supposedly, the availability to the RAF during WW2 of tetra-ethyl lead, was one of the factors that won the battle of Britain. Tetra-ethyl lead, the blending component which is added to make leaded gasoline, was able to give the british aircraft engines the extra performance they needed to outfly the Luftwaffe.

This is as far as my knowledge goes... Hope it helps.

Yes, the allies steadily increased the octane rating of their avgas during the war. At the end of the war they were using 115/145 (lean/rich) octane avgas. Or to be pedantic, when the number is >100, it's called "performance number" instead of octane, due to a different way to determine it (basically % power increase vs. 100 octane).

These high ratings were achieved mostly by ridiculous amounts of TEL.