Well you're not really lucky to be born in this century, since technically you could be born in 1999 and enjoy the same, if not more benefits. You could be lucky to live in the 21st century, though. Anyway- here it goes.
The first 14 years of the 21st century have marked the completion of the Human Genome Project, a rise in the global economy, the exponential development of technology and thus more efficient agricultural and industrial practices. One square inch of hard drive can now store well over hundreds of gigabytes, when even at the end of the 20th century, it could only store a few megabytes. Facebook, Twitter, and YouTube could only be in one's dreams of until this century. Consumer Protection Laws have been enforced more and me getting sick is less of a problem due to colossal advancements in medicine. Public smoking bans make the air cleaner Furthermore, learning has become so much easier and cheaper: I can't imagine my grades if I lived in 1984 and couldn't look up things to cram for a test.
No one can argue that there hasn't been revolutionary development in nearly every field during these few years but we've had a lot of not-so-good things happen as well. The Iraq invasion and Afghanistan War, pollution, and deforestation the rise among others. These cannot be ignored, but they are certainly less potent than the development.
So despite our ups and downs, I can say that we've had, and are likely to continue to have a better century than the ones before. If you're not convinced, imagine your life in any century, with less or no freedom, so much more corruption by government and businesses and police and government brutality, and less or no medical aid, among others. I say without a doubt, that I'm incredibly lucky to be born in this century.