In 1843 Ada Lovelace published a set of notes appended to her translation of Menabrea’s account of Charles Babbage’s Analytical Engine. The final note, Note G, lays out a step-by-step procedure for the engine to compute the Bernoulli numbers.
This procedure is widely regarded as the first algorithm intended to be carried out by a machine, which is why Lovelace is often described as the first computer programmer. The claim should be stated as “widely regarded as” rather than as a settled fact: the Analytical Engine was never built, so the procedure was never run, and the question of historical firsts is genuinely debated. The underlying document, with the notes credited to “ADA AUGUSTA, COUNTESS OF LOVELACE,” is itself the primary source for the claim.