Curiosity is on the Martian surface and is equipped with a slew of laboratory equipment. What would Curiosity need to discover to prove there is or has been life on Mars? Would it have to find DNA (or its Martian equivalent)?
No. In fact, DNA would be lousy evidence of life on Mars, it would almost certainly be a contamination from Earth.
Lacking equipment, I’m not even sure how Curiosity would go about finding traces of DNA but assuming that it did (through some sort of chemical assay), it has no way of telling the origin of said DNA.
And even if the genetic material could be sequenced there are a number of scenarios. It could be very similar to what is found in extremophiles on Earth, in which case we wouldn’t necessarily know the origin (but due to the aforementioned contamination risk, the prior probability of it being Earth life would be very high). If it were completely different, that might be compelling enough to consider it extraterrestrial.
But maybe not: estimates for the number of existing species on Earth vary drastically. One thing is certain, though: we’ve only scratched the surface of prokaryotic life in particular. So if we find something new, that wouldn’t even be very surprising.
More interesting would be an alternative genetic material, neither DNA nor RNA. So far, we haven’t discovered anything like this on Earth (despite earlier hopes) so it would be more likely of extraterrestrial origin. But again, as the refuted study shows, we cannot entirely exclude the possibility of such life being found on Earth either.
All this is just to show that proving microbial life to be extraterrestrial is quite hard. It would be much easier if we were to find non-microbial life on Mars (i.e. something multicellular). However, I don’t know of any serious scientist who seriously entertains this possibility.
But the reason to prefer such a discovery is simple: the bigger the life form, the less likely that we haven’t found it yet on Earth, and the smaller its viable niche. So if we find multicellular life adapted to the surface of Mars there aren’t many places on Earth where it could have originated from, never mind the low chance of having survived the ride aboard the probe.
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There is a whole field of study revolving around these questions called astrobiology. It is assumed that extraterrestrial life forms will have some sort of information storage similar to DNA but that it will probably not be identical. Accessible water is considered the most important indicator for the potentiality of life. Oxygen or some kind of reducing element is also considered to be very important.
In the simplest sense, extraterrestrial life is life found beyond the Earth. Defining what life itself is notoriously difficult - perhaps the best is a system which is subject to Darwinian evolution?
However, the term "extraterrestrial life" does not necessarily imply that said life (if found) originated in situ. For example, if we do find (presumably microbial) life on Mars, I would not be completely surprised if it had a DNA/RNA (or PNA or similar) information system.
The amount of material exchange (due primarily to asteroid impacts) between the Earth and Mars is large - significant quantities of Martian rocks have been found on Earth, and the reverse is presumably also true. Earth's higher escape velocity would reduce the amount of material ejected from the surface, but Mars' thinner atmosphere would facilitate the successful "landing" of any ejected rock without being burned up on re-entry beforehand.
So, Mars may have been seeded by Earth life traveling on rocks ejected by impacts. Whether this life could adapt to extreme cold, saline regolith, ionising radiation, dessication etc. is another matter although extremophiles on Earth are seen to (individually) have some of these qualities (psychrophiles, halophiles, "radiophiles", etc. Deinococcus radiodurans actually does quite well in all these areas.
Alternatively, Earth life may have been seeded by Mars rocks traveling the other way. If Mars was warm and wet in the past as some advocate, this idea may not be so crazy.
Look for "panspermia" and "lithopanspermia" for more information.