A mountain rising from the ocean floor whose top does not break the surface of the ocean to form an island.

Seamounts are created in bunches; wherever there is one seamount, you are likely to find a hundred. Of course, the distinction between an island and a seamount is meaningful only to the life forms that want to live on islands. The distinction is irrelevant to the geological processes involved. These processes create mountains; some become islands and others do not.

Seamounts are generally volcanic in origin; although some (such as Loihi) are active volcanoes that are building themselves up from the ocean floor, most are extinct volcanoes that either never lasted long enough to break the surface, or had once been islands but have been completely eroded away, and/or drowned by sea level rising1.

Near the margins of an undersea subduction zone, a line of seamounts will form in much the same fashion that a line of volcanoes will form near a continental subduction zone2. Examples of this are the Aleutian Islands, the Solomon Islands, and the Aeolian Arc in the Tyrrhenian Sea.

Alternatively, as a tectonic plate drags itself over a hot spot, the hot spot will periodically eat holes through the plate and form a line of seamounts/islands. For example, the same hot spot that created the Hawaiian Islands also created the Emperor Seamounts. These stretch west from Midway Island, and then north, forming a chain of undersea mountains stretching all the way to the point where the Aleutian Islands meet the coast of Kamchatka.
1If an island or a continent has a coral reef, sometimes the reef can grow fast enough to keep up with sea level change, and an atoll results. The Great Barrier Reef is also an example of this. However, drowned coral reefs are numerous. The Emperor Seamounts contain many.
2For example, the Cascade Mountains on the west coast of North America.

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