HFS (HTTP File Server) is a file sharing tool and simple web server, which also happens to be tiny (559KB), portable and open source.
The program is surprisingly easy to use. Drag and drop the files you'd like to share onto the "Virtual File System" pane, click the "Server is currently OFF" status indicator to turn it on, and that's it, for local users at least. Give anyone on your network the IP system displayed by HFS, and if they put this into their browser then they should see and be able to download your files.
You want to share files over the internet? Clicking Menu > Self Test is the best place to start. HFS will scan your system, check for various issues, and report on whether you can do so immediately, or what you need to do to make this happen. If you're a networking beginner then this advice may not always be straightforward - we were told to ensure our router was "configured to forward port 1000", for instance - but still, by comparison with similar tools, it's fairly easy.
If you're more experienced, though, you'll appreciate the vast array of options available here. The folders and files you're sharing can be password-protected, for instance, or restricted to particular users. You might allow people to upload, as well as download files. Various speed limits and controls prevent the program from tying up all your bandwidth. And there's dynamic DNS support, a configurable HTML template, a custom scripting language, and lots of configuration settings to help get everything working properly.
HFS has its limitations. While the program can be used as a web server, say, to host a web site rather than just share files, it's not exactly packed with features. There's no CGI or PHP support, for instance - there are many more powerful free servers available.
If you only need the file sharing features, though, it's a very different story. HFS is compact, configurable, and so easy to use that you could be sharing files locally within a minute or two.