The British Open cut rule is not difficult to understand: The top 70 finishers after 36 holes make the cut. Everyone else doesn’t make the weekend. This cut rule was first used at the 1898 British Open.
British Open: briefly with two cuts
But there wasn’t always this one-cut system. In 1968, the Royal and Ancient Golf Club of St Andrews (R&A) installed two cuts: one cut after 36 holes, another cut after 54 holes. This cut rule lasted until 1985, during which time the top 80 players advanced after the first cut and the top 60 players advanced after the second.
In 1986, the cut rule at the British Open was changed back to the best 70 after 36 holes. It is still in place today.