Both VPNs and the Tor network can be used to protect your online traffic and others from getting your real identity.
Setting up a VPN on Android is a fairly simple process. Searching for free VPNs in the Play Store will yield hundreds of results. The one I use however, is called Hotspot Shield and will work on most as it has been upgraded to use SSL, which can bypass many filtering services.
The app is very user friendly and only has one big button in the middle. After clicking it for the first time, you will be greeted with a connection request dialog you must accept. After this, you will be connected to a random server around the world, often in the same country. To connect to other countries, you will need to pay for Elite, which will give you more options and servers you can choose.
There are many other VPN services available, paid or free. Each has pros and cons, but all aim to serve the same purpose.
Tor is very different from a VPN because it can be used to randomize your connection every few minutes. This is secure as well, but can be significantly slower because it travels through many servers, rather than just once.
Once you click start, you can enable Tor for just the browser, or run your whole device and its apps through it. This uses the VPN feature of your phone to work, but is very flexible due to the wide variety of options.
With root access, you can transparently proxy your whole phone, making it even more anonymous and hard to detect. Also if the network you are on or country you live in is censoring your connection, you can use various bridges and alternate paths to still connect to the network.
Both VPNs and Tor can be beneficial when you are using your phone on public networks or if you need to access blocked content or services. They serve as a way to battle internet censorship and can allow access to things only available in certain countries. Tor is also open source, meaning you can take a look at its code and can compile your own version of it to use.