Proxy servers are used to access Internet by routing your request to a proxy server instead of directly to the server with whom you are communicating with. The proxy server makes a request on your behalf and return the result of the same back to you. It helps in cases of Internet censorship when a particular website or web service is banned in your location, be it a country or an office.
What generally people do is to open a proxy site in which they type in the web address or URL they want to access and then the page is displayed in the browser by the proxy server and all the link on the page are converted so that further requests when a user clicks on a link are routed & served by the proxy server only. But some desktop applications can also make use of proxy servers by configuring the options to route their requests to a proxy server for which you need the IP Address and Port number of the proxy server and login credentials (username & password), in case the proxy server is a private server and not open for general public. Example: Web browsers, FTP clients, Torrent clients, Media players etc
And sometimes there is even a need of a country based proxy because services like Last.fm streams free internet radio to users from US, UK & Germany. There can be several examples.
Whenever you need a good and stable proxy server, you can get a list at HideMyAss.com. The page itself keeps on updating the status of proxy servers.
You can filer proxy server listing by based on their location i.e. country, the port they run on (some applications only accept a proxy on a particular port or don't let you change it)
How to find a good proxy server?
If you observe the list, there are a lot of options to choose from. Good thing, but with more options comes the problem of picking up a good one. So, I will give you a quick tip to pick up a fast & speedy proxy server.
If you see the list, there are two bars which show the health of the proxy server. First one shows the speed and the other shows the connection time. So only look for those which show green state and hence are a good option. But chose those which are not completely green but those which are a little less because they are less likely to get a stream of more users and should sustain for more time before they go busy and stop working for you, hence you will need to make less switches as compared to picking up which is already stuffed by users. Moreover some web services might put up a restriction on users who might be using their service from the same IP (proxy server).
And also I would recommend to avoid using proxy servers for transferring any sensitive data like your gmail accounts, bank accounts etc. You can never be sure what they can do (I mean they are technically feasible of saving what data you pass through them which means your login credentials are exposed. If you have any questions, feel free to give me a shout in the comments.
In case you are a Stumble Upon user, then here is a quick tip (probably fastest method) to find proxy servers.