Looking strictly at real estate, the British Empire seems to have been tops, at about one-quarter of the world’s land surface. (And more than one-quarter of its total population, but population would be a whole other post.) Actual numbers for the land area of the British Empire at its height seem to run from 11.4 to 12 million square miles; this seems rather lower than one-quarter of the world’s land area, and in fact different sources describe the Empire as “20% of the world’s land area”, which seems more accurate, or even “one-sixth”, although that last seems low.
Apart from this rather, ah, eccentric site it’s hard to find quantitative numbers on the extent of the Mongol domains. (This author, who seems rather partisan, claims the Mongols are the champs; he gives a rather high number for the British–almost 12.8 million square miles–but outdoes that by awarding the Mongols 13.7 million. [The author’s even higher figure of 14.5 million square miles can be safely ignored, since he’s including “potential territories” which the mighty Mongols could have conquered if they’d really wanted to. Hey, and the British could have whipped Belgium’s butt and taken over the whole Belgian Congo-Zaire-Congo (Kinshasa). Or maybe not. Who knows? It didn’t happen.] However, his derivation of even the lower number is suspect, since he includes the entire area of the former Soviet Union in the Mongol Empire, and every historical map I’ve ever seen shows large areas of Siberia and northern European Russia outside of the Mongol dominions. He also seems to have thrown in a bunch of stuff about Prohibition and the war on drugs in there, which I didn’t really follow.) Some real quick back-of-the-envelope calculations of my own with a National Geographic historical map showing both the boundaries of the Mongol Empire at its territorial height and modern national boundaries, together with the land area statistics from the World Almanac, seem to give the Mongols something under 11 million square miles. Most sources tend to agree that the Mongol Empire was “the largest empire until the British Empire in the 19th Century” or was “the largest land empire” (i.e., the largest contiguous block of territory).
One could also add that the British were capable of exerting significant economic and diplomatic influence even in areas where they had comparatively small formal holdings, like South America, whereas there were whole continents where no one had ever heard of Genghis or Kublai Khan. A sea-based empire is arguably more powerful in that respect than a land empire. It’s also worth noting that neither the British Empire nor the Mongol Empire lasted for very long after reaching their peak territorial extents; both began gradually breaking up into increasingly autonomous areas very soon afterwards.