A proxy changes where your traffic appears to come from. It does not, without more, make that traffic private.
A proxy server is an intermediary between your device and the service you are reaching. The service sees the proxy's address, not yours. The general description is in Wikipedia's article on proxy servers.
Whoever runs the proxy can see where you are going, and, if the connection is not encrypted, what you send. Free public proxy lists are the riskiest form of this: you have no idea who operates them or why.
A VPN encrypts traffic between your device and the VPN provider and routes everything through it. That hides your activity from the local network. It moves trust to the VPN operator. It does not remove the need for trust.
Encryption protects the contents. A proxy or VPN changes who can see the destination. Neither protects you from typing your details into the wrong site.