How was the Byzantine Empire important to us.

A subjective question, isn't it? ;) I take it you mean from the western perspective. Let's look at a not so obvious effect that the Byzantines had: A lot is made of Charles Martel's victory at the Battle of Tours and its impact on western civilization, but he defeated the Muslims at the very remotest limits of their sprawling empire. From the 7th century until its end in the 15th, the Byzantines ruled a relatively small thumb of land directly abutting the very heart of the Islamic Empire. Constantinople was twice besieged by the caliphate before the two powers settled into a centuries long tradition of yearly border raids! It took 800 years for the Islamic powers to conquer from the Taurus Mountains to Constantinople. It only took the Ottomans 76 years to reach Vienna (for the first time) from Constantinople. It's possible we'd be looking at a very different world from our own had Constantinople fallen instead in 674 or 717.

What did they do?

This is a huge question! They stopped Muslim expansion for centuries. Just before and after the fall of Constantinople, their scholars traveled to Italy and injected Greek learning into the Italian Renaissance. Their religious and cultural inheritance is shared across the Balkans and Russia today. There is a great deal to go into, so could you narrow your question down a bit and spare my poor fingers?

Were they Greek or Roman?

The FAQ has some discussions about this. But yes, the "Byzantines" understood themselves to be Romans (Romaioi in Greek) and that lived in the Roman Empire (Basileia Rhomaion in Greek). The term "Byzantine" was invented by a German historian shortly after Constantinople fell and it became a part of modern historiography from then on.

Their eastern neighbors agreed. When the Seljuks conquered Anatolia in the 11th century, their rulers called themselves the Sultans of Rum, or Rome, because they had conquered much of what remained of the Roman Empire. And when the Ottomans conquered the last Byzantine territories in the Balkans in the 14-15th centuries, they called the area Rumelia because that was the "land of the Romans" from their perspective. The Latin west was dismissive of the claim, though, especially after the Charlemagne revived an empire in the west. Byzantines were very sensitive towards Latin snubs, which could lead to diplomatic incidents (at least until the Byzantines were too weak to press the point).