Yes and no. It's about as private as an email address. Everyone can always see which email address send a message (or which account purchased something with which bitcoin), but it's not easily traceable back to a real person. Then again if you're actually buying physical things with bitcoin then you probably had to supply a shipping address, so that's pretty easily traceable right there. Or they'd have to trace your internet traffic back to your computer. If they really want to find you they can.
The big draws to bitcoin were that:
1) It's not controlled by a government (so they can't intentionally inflate it)
2) your 'account' is not controlled by a third party (like a bank) which makes it harder for governments to shut down illegal activities because instead of simply requiring that the banks freeze all payments to $website they actually have to track down and shut off the physical servers in question (and for the more paranoid, it's also impossible for the government to simply freeze / steal your money, a la Cyprus)
3) there are no chargebacks, a spent bitcoin stays spent, so you can't buy something, wait for it to be shipped, and then dispute the charge like you can for credit cards (making bitcoins safer for merchants)
4) bitcoin transactions don't require any secret information (like for example your credit card number), making them safer for customers because you don't have to worry about identity theft
5) nerds liked that you could mine it using your spare cpu cycles and make real money out of nothing more than electricity (not true anymore, the market's been flooded with mining rigs and you are extremely unlikely to make more bitcoin than you're paying in electricity unless you get free electricity).