What is the name of the tallest mountain in the world?

As far as mountain concerned, the current convention is that “highest” means measured from the sea level to summit: “tallest” means measured from the bottom of the mountain to the top.

http://geology.com/records/highest-mountain-in-the-world.shtml

Mount Everest is the highest mountain in the world but it’s not the tallest.

Mauna Kea is the highest point on the island of Hawaii. The inactive volcano is a modest 4206 m above the sea level, but when measured from the seabed to its summit. It is 10, 200 m about three quarters of a mile taller than Mount Everest.

Reference:

 Mount Everest is Not The Tallest Mountain - KnowledgeNuts. (n.d.). Retrieved May 6, 2015, from http://knowledgenuts.com/2013/07/21/mount-everest-is-not-the-tallest-mountain/