A virtual private network (VPN) is a group of computers connected remotely to share files and resources, most often through encrypted tunnels -- especially useful when VPNs connect over untrusted Internet connections. This allows for the same network sharing as a group of PCs over a local area network (LAN) while using private key "handshakes" between users and servers to authenticate sessions before encrypting communication channels. A VPN does not change the appearance of your browsing sessions -- unless the service is advertisement-based, but we'll get to that.
VPNs were most often used in the corporate realm to securely connect employees to business resources, files and networks. However, consumers wishing to improve their online privacy are more and more often turning to VPNs as a way to shield their online activities.
As they have become easy to setup -- and often cheap to subscribe to, if not free -- accessibility and popularity has increased. They are often used to avoid censorship barriers and website restrictions in surveillance-heavy countries not only in the West but in areas such as China.
VPNs are only one piece of the puzzle when it comes to protecting yourself online. You should still use antivirus software and HTTPS whenever possible, and do your research before signing up -- some VPN providers will log your data (which removes the entire point of an anonymizing service) and with any online service, there may be security vulnerabilities which place you at risk. Keep an eye on your provider and any news related to these types of issues.
The most important reason is for the security and encryption added to your Internet connection and the web. Any data you send or receive is encrypted, making it a lot harder for anyone watching your activity to track you and what you visit.
In addition, a VPN can change where you appear to be logging in from -- removing blocks to geolocation-restricted content for you, such as a UK citizen who pays for a TV license wishing to access the BBC abroad, for example. If you're using a public Wi-Fi network, a VPN is a must to protect yourself and your connection.
Free and subscription models are available, but what's the difference? It depends on the provider, but "freemium" models -- offering a free service in return for displaying adverts or data collection -- is common. If you're happy to subscribe -- which can be anything from a few dollars a month upwards -- there are many options on the market. After testing a bunch of VPN services, I would personally recommend TorGuard for downloading and CyberGhost as a free option. If you're interested in researching others, click here for an extensive TorrentFreak list of VPN services and how they operate.