Woods was born in Dublin in 1903 and educated at The High School, Dublin. His father used a Harley-Davidson motorcycle commercially as a salesman for Mackintosh toffee, and Woods began racing in 1921 on the same machine. He watched the Isle of Man TT races at Hillberry Corner during the 1921 event and told his friends that he could do that, then set about proving it immediately.
Having no motorcycle of his own for the 1922 TT, Woods wrote to British motorcycle manufacturers seeking a ride. The Cotton company agreed to supply him a machine with a new overhead-valve Blackburne engine for the Junior TT. On first meeting the young Irishman, Cotton's racing manager reportedly exclaimed: "My God! They've sent me a bloody schoolboy!" The Isle of Man Examiner described Woods as an "enthusiastic amateur." He started number 40.
Woods's 1922 TT debut set the template for a career defined by tenacity. Despite knocking spark plugs from his pocket at the start, clipping a kerb at Governor's Bridge on the first lap, sliding off, having a pit fire extinguished with an overcoat, losing his brakes after the rear brake cam lever split in the fire, and falling again at Ramsey Hairpin on the final lap, he still finished fifth. He won the Junior TT the following year, 1923.
His association with Norton ran from 1926 to 1934, producing four Grand Prix victories in 1927, two more in 1928, and further wins in subsequent seasons. He became disillusioned with Norton machinery and in 1935 joined Moto Guzzi, giving the Italian manufacturer their first TT victory — only the second time in the event's history that a non-English motorcycle had won, after Oliver Godfrey's Indian in 1911.
Over the course of his TT career, Woods accumulated ten victories, a record that stood until Mike Hailwood surpassed it with fourteen wins. Three riders would go on to equal Woods's ten, while others extended the count further; the all-time record now belongs to Michael Dunlop with 29 TT wins.
The competitive landscape Woods navigated included battles with Jimmie Guthrie, Jimmie Simpson, Charlie Dodson, Harold Daniell, Freddie Frith, and Wal Handley. His riding style was influenced by watching fellow competitor Alec Bennett. He was known for bringing boxes of toffee to the Isle of Man for the Scouts who manned the scoreboards on which the grandstand audiences relied.
Woods won the Dutch TT six times on the early Assen circuit, and his 29 Grand Prix victories spanned multiple nations and multiple machines across more than fifteen seasons of top-level competition.
Woods was also an accomplished trials rider, competing into the 1940s. During the Second World War he served as Commandant and is credited with training some of the first Irish Army Motor Squadron personnel as the presidential Escort of Honour.
In 1957 he returned to the Isle of Man to celebrate the TT's Golden Jubilee, lapping the course on a 350cc Moto Guzzi at just over 82 mph. In 1968, a panel of experts convened to assess the TT's history named him the greatest of all the circuit's competitors. In 1996, the Irish Post Office issued a set of postage stamps featuring notable Irish motorcyclists, including a stamp depicting Woods.
He served as president of the TT Riders Association, maintaining a connection to the sport's community long after his racing career ended. Woods died on 28 July 1993 at the age of 89, having outlived most of the contemporaries with whom he had raced seven decades earlier.