The Tor project is free software that helps protect your privacy by making it difficult for a 3rd party to analyze your network requests or link your traffic back to your network access point. See the Tor overview page for reasons why this may be important to world citizens, corporations, or specific professions.
Simplified, this is done by using a large pool of distributed hosts and using varied and encrypted paths through these hosts to deliver your original request.
Be aware that no one is saying Tor provides fullproof anonymity on the internet, there are documented weaknesses [1,2,3]. But by now, it should be clear the security exists on a spectrum and not in absolute terms.
I will detail how to install both the Tor service and Tor browser which is designed to address the most common threats to remaining anonymous while browsing.