Each end (satellite and ground station) point a laser at each other, which they modulate according to the data they wish to send. The other end has a sensor (much like in a camera) that captures the light, and processes it to recover the data (demodulate). It's much like fiber optics, except over air instead of a cable.

How much power would such a laser emitter require? That is a huge distance to cover. Wouldn't it get attenuated over the distance?

Lasers are subject to the same inverse squared law as regular radio waves. However with lasers you're starting with a smaller beam so the attentuation is actually less