Submarine mountains form in a couple of main ways. They happen to be the same principles that guide the formation of continent based mountains as well. tectonic plate movement and a combination of the two are the reasons mountains form under the seas.
Tectonic drift is a major factor in forming these sub-marine mountains. The continental plates are in constant motion. Some moving together, others moving apart with most moving only about 1 or 2 centimetres a year. This in the short term has very little if any directly measurable effect on the formation of mountains. The main way which tectonics plays a role in this is when two continental plates collide. This causes either one shelf to rise and the other to slip under it, or both to rise until reaches a point of critical friction and slips, fractures and then what is left is a mountain or mountain range. The mid-Atlantic ridge is a prime example of what is possible when two tectonic plates collide head on.
Volcanism is sometimes also a by-product of tectonic movement but other times it has its own causes. Such as magma chambers deep below the surface of the crust that if unvented, basically act like a tea kettle. The pressure builds and tries to find a way to escape the chamber. It will do this by pushing through fractures in the bedrock, popping the area above the chamber like a cork or just plain exploding. Now once the magma reaches the icy waters at the bottom of the seas, it cools rapidly which causes the gases in it to create steam, air bubbles and fractures in the cooling lava-bed. This in turn allows the magma that follows it to seep a further out of the “Hawaiian Islands are a prime example of a submarine mountain that formed in this manner. Just to clarify, the only reason we call the Hawaiian Islands “islands” is because they rose tall enough to breach the surface of the water. It is still considered a mountain as such though.
These processes typically take many thousands, if not millions of years to complete a mountain. But they are ongoing and so long as the core of the planet Earth is a ball of molten Iron and Nickel, they will continue to shape the seafloors in ways that many of us will never see. Mother nature is a powerful force, one that does not act on a human time scale and one that we cannot predict. These mountains form all of the time. We just don’t really get the chance to see them.