Lets start from general meaning of word 'proxy'. Its actually an agent/subsitute or anybody acting at palce of any other.
Technically, proxies are used to bypass some firewall restrictions, maintaining anonymity or for many other purposes. Like you must using proxies for accessing social networking sites from your college network which your firewll restricts.
Lets understand the common methodology (talking in general terms)
You bunk the lecture, requests your friend to speak up your attendence, you get the attendance (in case professor doesnt' detect).
Now talking technically , when college firewall doesn't allow you to connect to a site (say facebook)
You requests a proxy server to bring the web page from facebook and serve you, Now the firewall thinks you are dealing with a server other then facebook and allows you to make a connection(in case firewall doesn't detect ).
Hence you have successfully managed to bypass a firewall. Proxy servers simply act as intermediate between your machine and actual server that you are accessing.
Proxy servers are mostly used to maintain anonymity. Suppose you are planning to commit an online crime,
you can use proxy server. The actual webserver doesnt come to know about you because proxy server is dealing with webserver on your beahalf.
Common ways to use proxies
1.Setting in Web Browser
1. First of all log on to whatismyip.com and pen down your current IP.
2. Logon to google.com and search for 'fresh proxies' You will get many sites providing you list of IPs and respective port numbers. It must be like this IP:port.
Suppose logon to http://www.digitalcybersoft.com/ProxyList/fresh-proxy-list.shtml
3. Now copy the IP and port no.
4. In Mozilla Firefox browser, go to Tools--Options--Advanced--Network--Settings . You will get the dialogbox as shown.
Check the option "Manual Proxy Configuration" , fill IP and port no. You can configure the proxies in any browser.
5. Check out whatismyip.com again , hopefully it mush have been changed. Similary you can configure proxies in other web browsers.
2.Using softwares. You can download many IP hiding softwares. There are easy to use and freely avaialable on internet.They automatically keep changing your IP after a particular interval of time. I would recommend you one that is Ultrasoft.
3. Using websites . Many websites (e.g hidemyass.com ) provide free services to hide you IP address.
You can directly visit other websites through them.
Note: If you are really conerned regarding your anonymity, do not use proxies. Read my tut on TOR here.