I was going to say!
No, we can't explain it concisely, because it's a massive piece of legislation, that does LOTS of things. It's also called the "Affordable Care Act" -- Obamacare is more of a nickname.
It was designed to enable most of the Americans who don't have access to health care to get it; and to prevent costs from continuing to sky-rocket, as they have been for decades.
A few of the big things it does: prevents insurance companies from refusing to offer insurance to people for "pre-existing conditions" (which means having ever had any medical problem or condition); makes insurance companies that spend less than 80% of the money they collect on actual health care (they can't keep 30% of premiums as profit, or waste it in "administrative fees"); and it sets up ways that people can buy health insurance, actually requiring most people to have at least basic coverage.
It does lots and lots of other things, but too many to list here.
Most Americans had been BEGGING for health care reform for decades. We actually wanted a single-payer system, as civilized countries have, but that wasn't considered politically feasible. While it was being proposed and passed, and since, Republicans have told innumerable lies about it, and tried to get it reversed, or otherwise tried to stop it going into effect (more and more provisions have been going into effect, with the biggies going into effect next year).
Edit: The "I was going to say" was "until I read you live in Europe" -- as it would be really strange to not have heard a lot about it here.
I wanted to correct the idea that it is "state paid" -- mostly it isn't. There's an expansion of the government paid system, but it won't cover everyone.
Also, the idea our health care system was ONLY considered broken after the recession is wrong. It's been a problem for a long time. But the Republican Party is never happy at anything that prevents or reduces needless suffering and death, so they have always hated the idea of making things better.