Largely we don't know. That's what makes them cool. This opens up a fairly large debate about where you draw the boundary of "life", "organisms", and "entities". It's a fun debate to have over a few beers since there is no objective correct answer.
My view is that Obelisks are more like Viruses or Viroids, or some kind of mobile genetic element. The key detail is that they appear to be strictly RNA elements (they don't have a DNA counterpart). So they're most likely using host RNA transcription machinery to make more copies of themselves, this is what viroids and satellite viruses like "Hepatitis Delta Virus" do.
What do they do? Well that's the right question. My guess is the kinds of things that bacteriophages do, Obelisks do too. Exploit cells to make more copies of themselves as selfish replicators.