Well, that depends...
Are you talking from the sea floor or from sea level?
If you are measuring from the sea level, then the tallest mountain in the world would be the famed Mt. Everest which is part of the Himalaya mountain range along the border of Nepal and Tibet in Asia, with the highest 'altitude' of 29,035 feet or 8,850 meters!
However, if you are measuring from the sea floor, (where the mountain originates from below the sea), then Mauna Kea, in Hawaii, beats Mt. Everest hands down!
But because a lot of Mauna Kea rests below the water, its height from sea level (where we can see it or begin to climb it) is less than Mount Everest: Only 13,796 feet.
Mauna Kea is over 10,000 meters tall compared to 8,848 meters for Mount Everest - making it the world's tallest mountain.
More Interesting Heights:
Highest Mountain in AfricaKilimanjaro, Tanzania: 19,340 feet / 5895 meters
Vinson Massif: 16,066 feet / 4897 meters
Highest Mountain in Australia
Kosciusko: 7310 feet / 2228 meters
Highest Mountain in Europe
Elbrus, Russia (Caucasus): 18,510 feet / 5642 meters
Highest Mountain in Western Europe
Mont Blanc, France-Italy: 15,771 feet / 4807 meters
Highest Mountain in Oceania
Puncak Jaya, New Guinea: 16,535 feet / 5040 meters
Highest Mountain in North America
McKinley (Denali), Alaska: 20,320 feet / 6194 meters
Highest Mountain in the 48 Contiguous United States
Whitney, California: 14,494 feet / 4418 meters
Highest Mountain in South America
Aconcagua, Argentina: 22,834 feet / 6960 meters
Source(s): teameverest03.org/, wikipedia, geology.com/, whyzz.com/, about.com/