I have learned that all currently-living organisms come from a common ancestor, which I theoretically understand. However, my professor in a class mentioned that there is a chance that there were multiple origins of life, but that all the creatures living today all came from the same one and the others died. How can we be sure that before the other origins of life died, they didn't somewhat evolve, and the species from that origin didn't just evolve in convergence with those of this one since through natural selection both would need similar characteristics to survive on Earth?
It is interesting that all forms of life use the same genetic code. The nucleotides composing DNA and RNA are universal. Eric Lander, geneticist from Harvard/MIT cites this as evidence for evolution from a single ancestor. I think your instructor may be saying that it's possible alternative adaptations for initiation of life may have occured and perhaps not worked out. They become just another of nature's experiments that ended in extinction. I seem to recall reading somewhere all but 2% of species have ended in extinction.
I believe that the answer to this question would be that we cannot tell for certain that there were multiple origins of life.
In order to determine that there were multiple origins of life, we would almost certainly need to look into the molecular biology side of these organisms, since convergent evolution makes it extremely likely that these differently originated organisms would be very close to each other in outward appearance. However, the basic molecular machinery would have much less of such selection pressures, allowing them to be maintained.
However, this would be impossible if the organisms have already gone extinct, as fossilisation generally does not preserve the structures of the proteins and DNA (or equivalent information storage molecules) that are used by the variant organisms.
If such an animal was alive, however, it would then be possible to determine the animals' variant origins by detecting a specific biochemical trait that they alone possess. For example, an animal which used D-amino acids for its proteins would certainly be determined to have originated from a different common ancestor.