Is this the first example of non-carbon-based life?
No. Carbon is still in the leading role, along with oxygen, hydrogen, and nitrogen. The claim here is that one of the important supporting actors - phosphorus - has been taken out and replaced with arsenic, an unusual agent to be sure.
In all life as we know it, phosphorus atoms make up important working parts of cells, including the so-called backbone of DNA - the part that holds the genetic code together. The NASA scientists say these bugs are using DNA with an arsenic backbone instead of a phosphorus one.
Do these bugs use arsenic in their natural state?
That's not clear yet. They were able to survive in a lake that had lots of arsenic, but the scientists added more arsenic in the lab. When the arsenic is removed, the bugs go back to using phosphorus, which they seem to prefer.
Why did they choose arsenic? Why not something more benign?
Arsenic is the element of choice here because it's like an oversized version of phosphorus. Arsenic and phosphorus form similar bonds with oxygen and other atoms, leading to molecules or parts of molecules with roughly the same shape.
In the periodic table, arsenic is right below phosphorus, and this is important because elements in the same vertical column of the periodic table share bonding similarities.
Silicon, for example, is right below carbon, which is why science fiction writers are fond of speculating about silicon-based life.
If you think of these atoms as Tinkertoy connectors or Legos, arsenic and phosphorus would make the same connections. But arsenic is bigger and often acts differently.
That's what makes it such a deadly poison to most organisms - it fits in the spaces meant for phosphorus, but once there it alters the way the cellular machinery works.