Proxy and VPN
A proxy is a server that routes connections in order to provide some additional service such as security, anonymity, or information processing. A common use is to connect to a proxy in order to mask one's IP address from a web server. There are thousands of proxy servers available, but many of them may not be trustworthy. One of the most secure system of proxies is Tor, a protocol for onion routing, the tor proxy is made up of three different proxy servers located in different countries, in order for an attacker to find out who is behind a website request he would have to get access to all servers and get hold of the logs, which are not meant to be kept by tor operators.
VPN
A VPN is a single hop proxy that is considerably faster than tor and that is its main advantadge, however a VPN is not to be trusted for high security and there has been instances in which people using a VPN to commit a crime have been arrested, when choosing a VPN, to minimize the risk of your privacy being compromised, one should look a company that does not keep connection logs for too many days and it is located offshore (i.e. not in your country of residence). Nearly all VPNs claim not to keep logs about the websites you visit but they do not specify anything about connection logs, it is not necessary to know what websites you visit to track you down, only connection logs, time and date of connection and what IP was assigned, are needed.
You can find free VPN services, they can be used to get around Internet filters and light privacy but not for high security, free VPN services are financed either selling your data to advertisements or giving you a limited bandwidth amount and hope that the user will upgrade to the paid for version.
VPN vulnerabilities
A very common VPN vulnerability is that the server sometimes will disconnect and show your real IP, this will be unexpected and there is little protection against it. One rough trick you can use to notice a VPN disconnection is to stream online music while you surf the Internet, for example listening to Pandora radio, when the VPN disconnects the music will stop playing, another way to stop exposing your IP is choosing a company that has VPN software with IP binding, Security Kiss has such feature, when IP binding has been activated it will stop your IP leaking out.
A second VPN vulnerability is a DNS leak, before you can access a website its URL needs to be resolved, DNS is used for this, by default you will be using your ISP DNS servers to resolve addresses, if your DNS leaks it would be possible to find your Internet provider by looking at that DNS name, although your computer IP would still be hidden. DNS leaks are sometimes patched by your own VPN provider but not always, if you use a VPN you can check if the DNS leaks visiting the DNSleaktest in external links. You can manually prevent DNS leaking changing your DNS resolver to a free one like Comodo Secure DNS, OpenDNS is another free DNS provider.
To completely avoid VPN disconnections revealing your computer IP the best one can do is to use an SSH tunnel instead of a VPN, unlike VPNs, SSH tunnels do not route all of your Internet traffic, they work on per application bases, when an SSH tunnel goes down your browser will not work and your real IP will remain secret until the tunnel is activated again. Privacy trends point towards VPNs, there are very few commercial SSH tunnel providers, some of them are VPNSecure.me, Cotse.net and Tunnelr.com
