Depending on exactly what you mean by "measured," there are two different answers to the fastest wind speed ever recorded on Earth, and both belong to genuinely extraordinary storms. The distinction between them comes down to how the measurement was actually taken, and it's worth understanding both.
The tornado record: 517 km/h, estimated by radar
On 3 May 1999, an F5 tornado tore through Bridge Creek, Moore, and the southern suburbs of Oklahoma City over the course of one hour and 25 minutes, travelling 61km across Grady, McClain, Cleveland and Oklahoma counties. It killed 36 people directly, injured more than 580, and caused around a billion dollars in damage in 1999 terms.
It also produced the highest wind speed ever measured on the planet. A team from the University of Oklahoma, using a mobile "Doppler on Wheels" research radar, measured winds inside the tornado at approximately 517 km/h, roughly 321 mph, at a height of around 32 metres above the ground. That figure comes with a real margin of error, plus or minus about 32 km/h, because it's an estimate derived from radar scanning the air inside a violently rotating funnel, not a direct surface reading from an instrument sitting in the wind. It remains, by a wide margin, the highest wind speed ever measured for any storm on Earth.
The surface wind record: 408 km/h, measured directly
The second record belongs to a completely different kind of storm. Severe Tropical Cyclone Olivia formed off northwest Australia in early April 1996 and, on 10 April, passed directly over Barrow Island, a small island off the Pilbara coast of Western Australia used for energy operations. An automatic weather station on the island, fitted with a physical anemometer, recorded a gust of 408 km/h, about 253 mph, during the eyewall passage.
Unlike the Oklahoma tornado figure, this was a direct surface measurement from an actual instrument, not a radar-based estimate, which is why it's recognised separately as the highest wind speed ever directly measured at ground level. The result was so extreme that the World Meteorological Organization didn't officially ratify it until 2010, 14 years after the fact, after a dedicated review panel confirmed the anemometer had been functioning correctly and that a gust of that speed was physically plausible for that storm.
Two different kinds of record
The reason both figures still stand as separate records rather than one simply replacing the other comes down to measurement method. Tornado winds are almost never captured by a physical instrument, because few anemometers survive being hit directly by a violent tornado, and even fewer happen to be positioned in exactly the right spot when one passes. Radar is how tornado wind speeds actually get measured in practice, which is why the Oklahoma figure, despite its larger margin of error, is accepted as the highest wind speed ever recorded for a tornado specifically, while Cyclone Olivia's gust holds the record for the highest wind speed ever measured directly by an instrument at the surface, tornado or otherwise.
Both events share one thing in common worth noting: they're both outliers even within their own category of storm. Most tornadoes, including violent ones, never get caught by a research radar at exactly the right moment to record a peak reading, and most tropical cyclones never happen to pass directly over a functioning, well-sited weather station at their most intense point. Both of these records exist largely because of where the storm happened to be at its peak, which is part of why they've stood for so long: matching both the intensity and the measurement opportunity again is genuinely rare.