One of the big names in social bookmarking is Delicious (http://delicious.com). It's had some ups and downs recently (it started as an independent start-up, got bought by Yahoo, Yahoo decided to stop supporting it, and it has now been bought and is getting updated by an entirely different company. Yeah, it was confusing at the time, too.)
Here's what you see if you look at my professional Delicious account. (You may want to click on the image to see it a bit better.)
On the left, you can see when I added each link, and you can see what tags I gave it. On the right sidebar, you can see some basic info about me, and then a list of all the tags I've ever assigned anywhere.
So, how do I add to my links? Most of the social bookmarking sites have a little link you put in your bookmark toolbar. Here's mine - off to the right, you can see a link to delicious, and then to "my delicious". The first one is what I click to add a link. The second takes me to my account and shows me all the links I've saved previously.
When you click on the "add a link" bookmark, a little window will pop up. Most of the time, I just add tags, and don't comment, but commenting is handy if you're creating links that will be shared with other people. (You can share a note about why you like it, what to pay attention to, etc.) As you can see in the bottom left, you can add this to a larger group of links (a stack - more on that below) or make it private (so only you can see it.)
The former version of Delicious had a way to bundle tags into larger groups (so I'd have one for library.profession, technology.tools, teaching.and.learning, and so on, that would gather smaller tags into one group.) The new version of Delicious has "stacks" where you can pick and choose specific links to include (though they make it easy to select whole groups at a time. Below, you can see what a stack looks like when I've added some items.
Finally, it's also possible to follow someone else on Delicious - that means that every time they save a public bookmark, you get to see what they saved. In practice, while I think this is a sort of neat idea (and it can be a very cool collaborative tool for some kinds of projects), I've never used it extensively.
Pinboard: an alternative:
When the future of Delicious was looking pretty uncertain, people spent a lot of time exploring some other options. One popular choice that rose to the top turned out to be a site called Pinboard (http://pinboard.in). They describe themselves as " a bookmarking website for introverted people in a hurry."
Unlike Delicious (which is free), there's a one-time fee (which goes up very slightly with each account created - currently, it's about $9.55.) but it is in fact very quick, simple, and the creator and admins are very responsive to concerns. You can see what it looks like here. (I've got an account for some professional uses, but I use it more heavily on my personal account.)
And here's what adding a new link looks like.
Twitter has become a great way for some people to share links of interest - the problem is finding them later when you need them. (So a lot of people will save a link to something like their browser, a 'read later' application like Instapaper, or to a social bookmarking site.)
There are a variety of other social bookmarking sites - I've only scratched the surface here. Popular sites include Digg, Reddit, and StumbleUpon, though all three have much more of a social factor than the two I've discussed above. (Great if you're working with popular topics, but sometimes a little weird if your interests are less common.)