Let us suppose that Bob has a computer at his home and an Internet account with his local internet service provider. Say, SuperNet Inc.
When Bob accesses a website from his computer, say www.google.com, the following takes place (simplified):
Bob's computer asks SuperNet Inc's DNS server for IP address of Google's web server. DNS, or domain name system, is like an address book: look up 'www.google.com' and get 184.108.40.206.
SuperNet Inc's DNS server tells Bob's computer that 'www.google.com' indeed resolves to 220.127.116.11.
Bob's computer initiates a connection to another compute on the Internet with globally unique IP address 18.104.22.168.
Bob's computer may be anywhere in the world and is not directly connected to Google's web server in any way. Therefore, the connection is routed (relayed) through several intermediary routing servers on the Internet, including of course SuperNet Inc's router. These servers have nothing to do with Google but pass messages between Bob's computer and Google's web servers.
Similar process takes place for all other Internet communication, including communication with email servers, file servers, etc.
A proxy server is a server that can do something for you by proxy, or on your behalf. For example, if Bob configures his web browser to use a proxy server, then instead of initiating a connection to 22.214.171.124 when he tries to access 'www.google.com', his computer will instead initiate a connection to the proxy server and will ask it to fetch / retrieve the www.google.com web page on Bob's behalf. As a consequence Google's web server will receive a connection from a proxy server, not from Bob's computer.
Virtual Private Networks
Usually, all computers connected to the same network can communicate with each other directly. For example, at Bob's home his computer can connect to his brother's computer to copy files, and Bob can print files to his wireless printer.
Mostly for security reasons, computers do not allow connections to them from the Internet. You wouldn't want random strangers accessing your files from the Internet, would you?
A VPN enables a computer on one network to communicate with computers on another network. Additionally, a VPN protocol (language in which a VPN connection is established and maintained) usually encrypts all communication that takes place over the VPN.
In the scenario above where Bob accesses Google web server, SuperNet Inc's router and each and every other network relay can 'see' all communication between Bob's computer and Google. Similarly, even when Bob uses a proxy server, SuperNet Inc's router and each and ever other network relay can 'see' all communication between Bob's computer and the proxy server (including the relayed communication to Google). In the latter case, the only difference is that Google's web server is accessed by the proxy server, and not Bob's computer.
Using a VPN service for security / anonymity is similar to using a proxy server. However, in this scenario the VPN is used to protect all communication between Bob's computer and the VPN server. SuperNet Inc's router and other network relays 'see' encrypted communication only.