Original use of proxies
Web-proxies are a tool for organisations to reduce bandwidth usage.
When a browser tries to show you a web page it first needs to download it. If you quit the browser or go to a different web page and later return to the original web page it has to download it again. This is wasteful and slow.
To speed things up many modern browsers use something called a cache. They keep (parts of) recently viewed pages in the cache. When you visit a web page the browser will first check if it already has a copy of that page. If it has a up to data copy it will show you the page from the cache, which is much faster and makes it unnecessary to download the page again.
The latter is a nice side effect for organisations with limited bandwidth, or who pay for their bandwidth.
However if you download the page from different machines then you will not get a local cache hit. (E.g. Jane visits a page from her machine. The first time it will be downloaded, the second time she views the page it will already be cached. Then Joe does the same from his machine with the same result. The page has now been downloaded a total of two times (and viewed 4 times).
If you can move the caching part to a place which gets used by both Jane and Joe then it would only be downloaded one.
This is something which you do with proxies.
Instead of downloading information directly from the web your browser connects to a proxy end tells it 'can you please download that page for me and forward it to me'.
The second user which visits the page gets a local copy from the proxy, resulting in a nett speed gain for the average user and lower bandwidth usage for the corporation.
Since all request for page download now come from a single machine (the proxy server) it is hard to track who is looking at which page. This has the advantage of anonymising the requests
Since all request flow though a single server it is easy for the person controlling to server to manipulate it. (e.g. keep statistics, block some pages etc).
As to the questions:
- What proxies differ from socks ?
Socks is a way to talk through a proxy. Which makes the question similar to "What differs French from talking over the phone".
- Who are behind those public proxies?
Look at the webpages of the proxies. They will tell you.
- How people find them and share them online?
That totally depends on the people. Personally I would Google on "list of public proxies" which returns a lot of results.