麻 is short for 麻臉 which means pockmarked face, 婆 is short for 婆婆 which means old lady, and 豆腐 is tofu, so 麻婆豆腐 is pockmarked old lady tofu.
The dish is apparently named (affectionately, I assume) after an elderly woman who invented it in Sichuan in the 1800s, whose now-immortalized facial scars were due to childhood smallpox.
Mapo tofuis one of the most well known Sichuan dishes, soft tofu simmered in a spicy sauce with minced beef or pork.