Your cousin just announced the birth of his child, and you're the very first to leave a congratulatory comment. The excitement for your beloved family member is quickly replaced with a steady flow of beeps and dings, alerting you to each and every comment left on the same post.
Luckily, there's a way for you to leave your comment and then politely excuse yourself from the conversation, without anyone even knowing.
Update: Facebook has since rolled out a new notification box, complete with a shortcut to unfollow posts. The method explained below still works, but you can save yourself a lot of time by using the new method. Read about it here.
If you make a habit of viewing and commenting on posts strictly from your Facebook News Feed, or only on your mobile device, odds are you've never noticed an Unfollow Post link. But, it's there, and here's where you find it.
You will need to be browsing Facebook on your computer, not your mobile phone, to view the thread causing all of the alerts. By clicking on the time the post was made you will be taken to that particular post's page.
Or, when you get an alert for a new comment, you can also click on the alert to view the post's page.
Once you are viewing the post, unfollowing it is as simple as clicking on the Unfollow Post link. Going forward, you won't receive an alert when a new comment is left, hopefully helping you sleep a little better at night.
Should you change your mind later on, simply view the post again and click on the new follow link.
The same method can be used to follow a post on Facebook, even if you haven't left a comment. By viewing the post, you can then click on the Follow Post link, allowing you to keep tabs on future comments, without having to leave a comment.

One of the biggest reasons I don't leave too many comments on Facebook is due to the alerts afterwards. By unfollowing a post, I can now comment alert-free, and engage a bit more with my "friends."
Updated May 18, 2012, to reflect the new, shorter method to unfollow a thread.