How exactly is a Tsunami caused

Is it basically earthquakes happening under the oceans surface?

Yes.
Most tsunamis are caused by earthquakes generated in a subduction zone ( underground where plates collide ).

go here and it will tell you everything about it and the process of tsunami.

http://geology.com/articles/tsunami-geol…

Technically, yes. The plates that are essentially the Earth's crust move and the water in the ocean moves because of this. Think of a bathtub full of water, with your hand on the bottom of the tub, move your hand in a violent fashion one way of the other and the water will move..the same thing is happening in the ocean but on an astrnomical scale, which forms a tsunami

A tectonic plate can be pushed up from a earthquake ( a Descructive one) causing the water to be pushed up and creating a wave, you would think it would slow down before it would hit the shore, but it keeps getting bigger and faster.

WARNING: if the sea is out further than ever, and you can see a wave in the distance, dont just stand there and watch the tsunami come inland and crush you to death, shout "TSUNAMI" and run to higher ground.

after their is an earth quake the force of the tectonic plates not only cause the earth quake but make a big wave

by a earth quake with a mangetout of 5.5