Usenet was originally developed as a discussion system where people would 'post' to a group (similar to a threads in a forum). Now it is largely used for posting files. Unlike P2P the files are stored on a commercial server, giving far greater speeds. The range of files stored across Usenet is slightly less than what you might be able to find looking through P2P networks.
There are many different providers of Usenet access. There are cheaper services like AstraNews (~$10/month) and more expensive services like Giganews ($30/month). The main advantage of 'premium' providers is that files uploaded aren't deleted for much longer (the 'retention' period). Giganews hasn't deleted posts in ~4 years. More expensive providers also offer better speeds and higher 'completion' rates (the amount of data that isn't damaged). It is really a matter of finding what service suits your needs & budget.
There are a number of sites that allow you to search files uploaded to Usenet. You then choose what you want to download and a .NZB file is created. This is opened by a reader (similar to the way you download and open a .TORRENT file). There are a number of readers around, some paid, some free. Personally I use NewsBin Pro, which cost ~$30.
Most files on Usenet are in .RAR format, and are split into multiple segments. In order to compensate for the possibility of some the downloaded data/segments being missing/wrong most file uploaders also include parity data (as .PAR2 files). These check the downloaded data and, if damaged, use a mathematical process (called Reed-Solomon error correction) to repair the data. This is similar to the system used in RAID-5 to allow for one of a collection of hard drives to fail without losing any data. Most readers include the ability to perform these checks, and unpack the .RAR files automatic.
Another paid option is to use one of the premium web file hosts, such as Filesonic or Wupload, along with a file search engine. These sites normally allow free downloading, but with strict limitations (waiting times, having to enter a capacha, limits on the size of files you can download, limits on how much you can download etc...). There is a huge amount of data hosted on these premium services, comparable to Usenet. It is also generally easier to use. Files are often uploaded as-is, rather than in split RARs, and any web browser will download the files.
Currently I am using Giganews & Filesonic as my main file sources. About 90% of the files I want can be found on one of these services.