A VPN is a virtual private network. It connects you to a remote server and encrypts your traffic so that nobody can see what you are doing on the internet.
You might use it to access content which you wouldn’t otherwise be able to view from your location, for example due to censorship or geographical restrictions. You could also use it if there was some kind of leak on your computer or other device, such as email messages or passwords being transmitted in plain text (clear text), where these should be encrypted with secure protocols like TLS/SSL.
The most popular VPN protocol is OpenVPN, which uses a combination of symmetric-key encryption (meaning that the same key is used to encrypt and decrypt) with Rivest–Shamir–Adleman (RSA) public-key cryptography.
In this case your web traffic would be encrypted by using RSA, then decrypted at the other end.
A proxy server is a computer which allows you to get around restrictions and censorship. It does this by either masking your IP address (the unique number assigned to every device on the internet) or replacing it with that of another computer.
Usually, proxies are used to get around regional restrictions on content. For example, if you were in a country where certain websites or services were blocked by the government or otherwise censored due to their unacceptability under local laws.
You could use a proxy server with encrypted connections for those websites and any others which might be blocked.
Proxies are also used to hide your identity or location if you want to do things like avoid being tracked by advertisers.
They are used for a wide variety of purposes, including circumventing censorship, anonymity (where you don’t want people to know who you really are), and even hiding your IP address from the websites that you visit.
In a traditional proxy setup, the first computer (known as the client) connects to an intermediate server by passing along its IP address. The proxy server then fetches any content that is requested from elsewhere on the internet and sends it back to the client.
This means that if you are using a VPN or Proxy, your web traffic will be encrypted before it leaves your computer; anyone monitoring your traffic cannot see what websites you visit or anything else about what you’re doing online. This makes them very useful for people in countries with severe restrictions on freedom of speech.