Lots of question, but I'll try my best to answer. Before I do I'll say one of the best ways to understand rss and feeds is to start subscribing to some. That will help show how useful they can be.
Basically what's going on with a feed is a feed is published in a more standardized way then a web page. That standardization helps programs to understand your content better in order to present it. For example most html pages will have a title indicated by <title>Page Title Here</title>, but an html page doesn't need a title in order to be displayed in a browser.
The standardization is like an agreed upon contract. Any page published as a feed guarantees to include certain information so any program that wants to read the feed knows it's there and can account for it.
You could create your own feeds for all your content, but most applications today will create one for you based on the html you're already publishing. This forum publishes a feed, which is created automatically by the vBulletin software. WordPress is another popular application that will publish a feed of your content for you.
The other side of the coin to publishing a feed is reading a feed. Modern browser understand feeds and allow you to subscribe through the browser as a live bookmark. It's 'live' because as the more content is published the actual pages stored in the bookmark change to refelct the new content.
Browsers aren't the only applications that read feeds. There are many stand alone feed readers and many online services that will also read feeds.
Here's an example of how the whole thing works. I have a blog that uses WordPress. When I publish a new post WordPress sends out a ping. The ping alerts any program that has subscribed to my feed that I've published new content. The subscribed program can then grab a copy of my content to display.
You might subscribe to my content in a stand alone reader (also called feed aggregators) Here's a
list of feed aggregators. Say you're using FeedReader as your aggregator. You've subscribed to my blog and I've published something new. The next time you open FeedReader it will let you know I've published a new post and display it for you. You don't need to visit my site, but you can still read my content.
If you're only subscribed to a couple of feeds it might not be a big deal to visit the sites every day or week and read the content there. If you're subscribed to a lot of feeds it can save a considerable amount of time to open your feed reader and have it let you know which sites have new content.
When you read content from a web page you're going to the web page. With feeds the situation is reversed. Now the content is coming to you.
FeedBurner is a service that helps people subscribe to feeds. It is free as is Google Analytics. Even though feeds are standardized, there's more than one standard. Different applications may have different ways of subscribing and understanding your content. FeedBurner helps in the process of connecting readers and publishers. And for publishers they provide some stats on how many people are subscribing to your feed and what program or service people are using to subscribe.
If you burn your feed at FeedBurner then on your site you'd link to your FeedBurner feed to have people subscribe. When people click the link they'll be taken to FeedBurner and be able to subscribe through any chicklet they want.
You don't have to leave an email address to subscribe to a blog, though you can subscribe via email and have new content delivered to you as email. In that case you would naturally need to leave an email address.
As long as you're publishing a feed of your content you don't need to do anything special to tell people you have new content on that feed. However if you've been publishing a blog feed and later add a forum the two will have different feeds. Someone who subscribed to your blog won't be subscribed automatically to your forum. What you would probably do in that case is write a blog post letting people know you have a new forum and sharing the link to the forum or the forum feed within your blog post. Then anyone who reads your blog and wants to also subscribe to your forum will know about it.
In general the advantage of feeds is that readers can consume your content in a way that's more convenient to them. I'm subscribed to several hundred feeds. Way to many to visit each site every week. I open my feed reader (I use the sage too extension for Firefox) and the reader lets me know which feeds have new content. I read what I want and can easily ignore the rest.
The advantage for the published as that your content can reach more people. While may not be visiting all of those several hundred sites to consume their content, I am at least reading the content. Without a feed reader I wouldn't be reading most of them at all, since it just wouldn't be possible. More people reading your content means more people are talking about you. You have a larger part in the overall conversation on the web.
As another example if you look to the left just under my avatar you can see my latest blog post. That's possible because my the forum is set up to receive my feed and when I publish new content my blog alerts the forum that it has new content.
Again I'll recommend subscribing to a few feeds to see how it all works. You can grab one from the list I linked to (You possibly already have one or more of the programs listed), find a few sites that publish feeds that interest you and subscribe.
I've really only scratched the surface of rss and feeds so feel free to ask more questions if you have any. I'm sure I've forgotten to say a thing or two that you might want to know.
Hope something above helps.
