It's used to be pretty frequent in movies where I think the person after accomplishing a big task would want to convey to the viewers that "I am awesome and a hot-shot". It starts with bringing your fingers (curled towards the palm) close to your mouth and blowing air on the nails and then wiping something off the left side of chest with your fingers(or nails?). Does there exist a concise phrase I could write instead of describing the whole act? If this on topic, what was the origin of this gesture?
The gesture is polishing one's fingernails. You mimic blowing some moisture from your breath onto your polished fingernails and using that to restore the shine by rubbing them on your jacket or shirt front. It means, "That impressive thing I just did? I'm so good that it caused me no more trouble than dulling my manicure a bit."
Is this still a thing?
It's called buffing [one's] nails and might originate from the Bugs Bunny cartoon. Source
I'd suggest that there are two gestures here. The first one is buffing his or her nails.. and the second one is "wiping dirt off of your shoulder" or "Brushing it off" which kind of indicates that something is "no big deal" or even that they are better than the thing they are brushing off... its a bit of a showboat.