1.proxy means "something that acts on behalf of another thing." This is an English word, not an IT word. For example, proxy voting is when you ask someone else to vote for you, usually because you trust their decision.
In the case of a network proxy, it acts like a local web server and can request pages, but will fetch them from a real web server and then serve them to you. Similarly, in the case of the database server, the database proxy will execute the SQL query, send it to the remote database server, get the answer, and then pass it back to you. In code, a proxy can be an instance (or object) that runs on the local machine and receives commands, as in object-oriented programming, but actually passes those commands to a similar piece of code running on another machine.
Typically it caches pages in the process. Then, if it's dealing with the office, for example, you visit Microsoft.com and three other people visit Microsoft.com, the first request lets the agent go out and get the page, and then pulls subsequent requests from the cache, making them faster.
2. Proxies can also check to ensure that content does not contain malware and record your web browsing activity.
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