I often get asked about anonymous surfing and P2P (BitTorrent, eDonkey, DirectConnect, etc.). Even a less cautious user knows how much his/her IP address reveals. So how can you hide it?
Tor is an anonymous network based on P2P technology. However, it is not meant for P2P programs, as they generate too much traffic for the network to take at its current state. You should not route P2P traffic through it, go for an anonymous VPN service instead.
Tor is best used for anonymous internet surfing. Every request goes through several "hops", meaning your page queries and browsing could go through USA, Japan, Sweden or any other country, where every server node running a Tor proxy only sees the previous and next proxy, not where the query originated from. Tracing down a Tor user is extremely difficult, especially if the user does not run a Tor node him/herself. The traffic is also encrypted. I personally run a Tor exit node to donate traffic for the network.
Routing all traffic through a VPN connection is easy and there are hundreds of service providers on the internet that hide you behind their IP address, either for free or for a small fee. In this case the traffic travels only through one server, which maintains better speed than Tor. You can also run P2P programs over anonymous VPN services (unless it's forbidden in its terms) to hide your own IP address from prying eyes. I've used the following anonymous VPN services:
Free, but shows ads in your browser. You can eliminate the ads with Adblock Plus in Firefox, though. The service also disconnects you after some hours (from the server, config file change won't help), so it is not good for over-night use. There is a monthly bandwidth limit of 3 GB, which is tied to your network card's MAC address. Under OS X you can change this address from the Terminal with "sudo ifconfig en0 lladdr xx:xx:xx:xx:xx:xx" (replace every "xx" with a random hex number, eg. "00:01:02:03:04:05"), for other operating systems see here. The service is OpenVPN-based, which guarantees strong encryption. However, it does not work under Linux with plain OpenVPN + the config file, but requires its own program instead, which is a huge downside.
Not free, but very affordable. You can test the service for a few dollars per month, or even pay a whole year in advance. Stays on overnight, no bandwidth limit, no ads. You can choose between PPTP and OpenVPN. PPTP is supported out-of-the-box by pretty much every operating system, but OpenVPN traffic is harder to notice and doesn't contain the weaknesses in encryption that PPTP is vulnerable to. VforVPN promises not to keep any log files of their users' activities. This is a question of trust. The service sees the originating IP address, unlike in Tor network. You can only hope they are true to their word.
As a general rule, while the traffic is encrypted, you should still be aware that it travels through proxies that could work as middlemen to capture it. Therefore, don't send confidential data and avoid using your internet bank or PayPal through these anonymous services.