Concerns about internet privacy are no longer just the realm of child pornographers, terrorists, and hackers: a compromised internet identity makes you a prime target for identity theft, and other illegal activity. Some people are also worried about staying safe from their own governments (and reasonably so!). If you want to keep yourself safe in this digital age, you can do so by taking some basic precautions which serve to hide or disguise your identity.
To be anonymous online you first should understand, the basics for serving anonymity online.
Every time you visit a website, it will log your IP address (your computer's "address" on the internet), what site you're coming from, the browser you are using, your operating system, how long you spend on the site, and what links you click.
Major search engines store your search history. Your search engine queries are associated with your IP address (and account if you are logged in). These are compiled and analyzed to more accurately target ads and provide relevant search results.
Your ISP (internet service provider) can analyze network traffic to see what you are doing online. This is most commonly used to determine if customers are using the network to download torrents of copyrighted materials.
if your computer is logged into any social networks (Facebook, Twitter, etc.), these networks will be able to track your browsing history if the websites you visit have social network plugins ("Like" buttons, Retweets, etc.).
Complete anonymity is virtually impossible. No matter how much you cover your tracks, there is always some information that can potentially be used to profile and potentially identify you. The goal of using anonymity tools is to reduce the amount of information available, but due to the nature of the internet, you cannot ever be truly anonymous.
It's not exactly easy to stay anonymous online, and requires a significant amount of conscious effort. You'll face much slower connections when browsing websites, and you'll have to jump through more hoops before you even go online. If your anonymity is important to you, be prepared to make some sacrifices.