"I am a man."
Om means in most contexts human, but it can also be used to mean man, like homme or uomo. Bărbat is the word that means man, and you're going to use this in 99.9% of your conversations :) Also be careful with the spelling, it's not hard, these sounds are present in English too, so better have a good start ;)
Depending on how you use it. When you refer to it as an attribute you don't to use un, but when talking about a specifical thing/being you should use the indefinite article. Here he defines himself as a man, one specifical being, so un is used. Hope my advice was useful <3