P2P examples from my use

I don't know about a lot of the filesharing methods. The trick is finding what suits you personally and getting comfortable using those tools. No one can ever completely understand the infinite complexity of computer networking, but a working overall perspective isn't that hard. Look on wikipedia or infoanarchy or the millions of other internet forums for information. Unfortunately, jargon is necessary to explain these things, but the internet also hosts these instant dictionaries of the jargon.
I'm going to focus on a few specific technologies I have used as a kind of howto example. Sometimes there is too much choice and just getting started in anything decent and popular is the best introduction for a newby. Please note, software changes fast and I may change my mind in the future. Take it all with a grain of salt and make up your own damn mind!
Napster was the first most famous of the mass P2P era. Popularity and name recognition are significant with these things because they determine selection; the more users, the better library to draw from. KaZaa and WinMX came after Napster was shut down and featured more fundamentally decentralised technology. These programs have their own internal search functions, which is often the point at which centralised control and legal pressure can be applied. Many of the older networks are dead because the RIAA killed them in court. Currently, the program I use that is still working great is called Soulseek. It is an established system started in the KaZaa / WinMX era. It was not shut down mainly because it's used for trading independently produced, locally significant, obscure, and community types of music. One of the best features is the 'rooms' where you trade with users of similar tastes. I recommend the open source application Soulseek.
There are lots of great music download websites out there, or just sites which give you the titles of good music that then you can then plug into a P2P search. Some of the music sites I like host mp3s for centralised direct web download, because they are obscure bands in favour of filesharing. Of course, it all depends on your taste. I especially like the CiTR radio show pages for These are the Breaks and Radio Zero, Nardwuar's site, Electropunk hitlist, Evolution Control Committee, Negativland, Chumbawamba, Illegal Art, Shirley and Spinoza Internet Radio, People Like Us, wobbly, cutcopy, chicks on speed, peaches... I could go on an on.
BT = Bittorrent
BT is an interesting phenomenon that has taken the internet by storm and now, according to some reports, accounts for over 50% of internet traffic by volume.
It is not entirely decentralised, but it is a very functional form of P2P. To download a BT torrent, you must simultaneously upload a similar volume of data, even before the complete file is acquired. It was designed initially for releasing large open source operating systems, massive popular files moved with efficiency. In this type of P2P architecture, the more users you have, the faster the torrent will be shared. So, the issue of a popular file on a centralised server that gets overloaded is resolved. However, availability is heavily skewed towards only the most popular files.
One thing that BT lacks is anonymity. To download a file using BT, you first need a separate .torrent file of instructions that sends your computer to a centralised tracker to manages the sharing between you and the others. The .torrent files are usually posted publicly on a web site so they are easy to find and search using web searching techniques. Posting a .torrent that instructs your computer in downloading illegal mp3s is not currently outlawed in the way that posting Brittany Spears mp3s on your home page would be. However, the law is changing fast and these non-anonymous and centralised aspects of the system expose it to centralised failure via law. Rodi, as an encrypted P2P program that copies the mandatory sharing architecture of BT, but also hides the user so that the network cannot be shut down via centralised attack. However, this program is still in the primary stages of development, and I've never successfully used it to explain how it works.
torrent software clients
I recommend using Bittorrent [BT] technology to get big files such as movies or discographies [all the music published by a certain artist in one bundle]. Smaller, more obscure files are better shared with something like Soulseek. There are many different programs, called clients, that use the BT method. It doesn't matter which client you use, the network is limited by the information in the torrent file only.
Azureus is the most popular client for BT out there but is written in java, which crashes my computer. I use to use ABC because it is open source and written in Python which is better language than java. But then, I learned about µtorrent [aka utorrent because people don't know that if you hold down 'Alt' then numbers 0 1 8 1 you get 'µ'], which I now recommend. It is a very small program. You don't need to install it, you just run the .exe. It's very decent for windows and crashes a lot less than even ABC did. It doesn't slow down my computer unless I'm sharing a large volume of traffic. It is closed source, which sucks, but it's currently distributed free.
The other downside of utorrent is a problem with file size that ABC did better. This means that I run out of disk space artificially sometimes. However, utorrent is superior not just because it is a smaller .exe, but also because it has features like DHT which mean you can keep sharing torrents without a centralised tracker, meaning torrents which are dead on ABC may be alive and well on utorrent. This is the same feature as in Azureus.
For BT of any sort, you need to download the .torrent files themselves. I find mine using isohunt.com. It is a big commercial site with sketchy pop-up ads that would be dangerous to click on. However, isohunt has the best selection when I've used it because it metasearches all the other search engines. utorrent comes with a list of other torrent search engines. The only other ones I've tried are mininova.org and piratebay.
To find less popular material various topic specific websites with closed tracker communities are better - if you know what you are looking for. Such as: Southpark, the Simpsons, Electropunk music or indymedia torrents. There are many others. Sometimes you have to sign up with a tracker through their website in order to be able to download a private torrent they are hosting.
find more
I put GNU or GPL in with my search terms to filter out commercial software. Don't rely on Google ALL the time, monopolies don't tend to highlight the little guy. There is also dmoz, a people compiled search engine. I tend to rely on wikipedia a bit too much. Also, I think yahoo now enables Creative Commons specific searches. InfoAnarchy has a wiki all about filesharing.