How do VPN and proxy servers work?

You may not know it (and once you do not know it, it may not bother you), but every time you reach out to a website or connect with anyone online, your online connection gives your computer "address" to the site/person you're connecting with.
Why? So that you the other end knows how to send information (a Web page, email, etc.) back to your computer...to you. That address is your public IP address. IP stands for Internet Protocol.
Without an IP address, you wouldn't be able to do any Internet/online activity and others online wouldn't be able to reach you. It is how you connect to the world.
A VPN or proxy lets you go online under a different IP address identity.
You don't change your Internet provider; you simply go online and search for "free proxies" or "list of proxies" and you will get several websites that provide lists of free proxies or commonly known as VPN servers.
It's that simple to find servers, just like you order shoes, movies and airline tickets online. However, it isn't so simple to figure out how to use one without some guidance.
How a VPN or proxy operates.
A VPN server is a computer on the web that redirects your web browsing activity. Here's what that means.
Normally, when you type in a website name (Amazon.com or any other), your Internet Service Provider (ISP) makes the request for you and connects you with the destination—and reveals your real IP address, as mentioned before.
When you use a VPN your online requests get rerouted.
While using a VPN, your Internet request goes from your computer to your ISP as usual, but then gets sent to the proxy server, and then to the website/destination. Along the way, the VPN uses the IP address you chose in your setup, masking your real IP address.
Even though all proxies help you access websites you might not otherwise get to, not all proxies behave the same way. A VPN can fall into one of four categories:
Transparent proxy. It tells websites that it is a proxy server and it will pass along your IP address anyway.
Anonymous proxy. It will identify itself as a proxy, but it won't pass your IP address to the website.
Distorting proxy. It passes along an incorrect IP address for you while identifying itself as a proxy.
High Anonymity proxy. The proxy and your IP address stay a secret. The website just sees a random IP address connecting to it...that isn't yours.