The classic Mega Man games from Mega Man 1 through Mega Man 10 feature 78 Robot Masters in total. The count uses the original releases: six Robot Masters in Mega Man 1, followed by eight in each of Mega Man 2 through Mega Man 10.
That arithmetic is 6 + (9 × 8) = 78. The first game is the exception because it predates the series’ standard eight-boss format. Beginning with Mega Man 2, selecting and defeating eight themed Robot Masters became one of the franchise’s defining structures.
The number can become confusing when remakes, downloadable content, and spin-offs are included. Mega Man Powered Up adds Time Man and Oil Man to the first game’s roster, so a count that includes those extra bosses is not counting only the original Mega Man 1–10 lineup. Mega Man & Bass is also outside the range named in the question.
Robot Masters are generally humanoid robots built for specialized work or combat, and defeating one normally lets Mega Man copy its signature weapon. That weapon-based structure is why the bosses are counted separately from fortress bosses such as the Yellow Devil.