I discussed in another post the negative effects that blogging can have on your personal life and career and the importance of boundaries. One way that some bloggers prevent these dangers and others (such as stalkers, threats, or identity theft) is to remain completely anonymous.
It's certainly easy enough to do. Don't mention your real name, where you live, or too many details about yourself. Create an e-mail that you only use for blogging and uses a nickname. Make sure the anonymous blog is not linked to your other social media that may reveal more about you.
Blogging without revealing who you are has a lot of benefits. You can say just about anything you want (within reason) without damaging your career or personal relationships. This allows you to express views you may have but don't normally share and take on controversial topics. It also means you do not have to share your blogging habit with anyone who might not approve.
On the other hand, people may take you less seriously if you take advantage of the anonymity that the Internet offers you. You will most likely be challenged for it. Also, it makes writing personal blogs more difficult because not only do you have to change names, you will have to omit a lot of information that might reveal your location or identity.
Either way, it is a judgement call of how much you are willing to reveal. Plenty of bloggers do use their full name, but these are mostly professional bloggers. The best thing to do might be a balance of the two. Use a fake name, but for everything else use general information.