As with many things, it's been long enough that Extensions have come back around. Like Star Wars and Mad Max films, it seems that since enough time has passed, we get a new version of something we remember from when we were young.
If you're a more recent convert to Apple's operating systems, your first exposure to Extensions was probably in iOS 8, where extensions show up as options you can use to customize the "Today" screen, or as other choices available when you tap an action button, such as "Open In…"
Luckily these are not the same sort of extensions that drove many a Mac admin to drink in the 1990s. Thanks to sandboxing and different architectures in Mac OS X, the dark art of Extension Management is much simpler now. All you have to do is launch System Preferences and go to System preferences > Extensions and you can see all your extensions.
Yosemite's Extensions panel listing all available. Your mileage may vary.
If you're looking for a specific extension, or just want to know about a particular type of extension, you can click on Actions, Finder, Share Menu, or Today. Each of these groups only shows the extensions in each category, and you can view all available extensions at the top under "All." This view could be convenient if you have apps installed that might have more than one extension, since maybe those extensions do different things and now you can see all of them in one place, sorted by application.
Actions are for content viewing and editing, Finder extensions are used to enhance the Finder. Share Menu extensions offer a variety of ways to share content right from the app you're using (want to send a birthday greeting to someone via text/email/Twitter/Facebook? Done!) and is most likely to be the longest list of available extensions. Today is a list that shows the Notification Center widgets on your Mac.
Yosemite's list of Sharing extensions. No 50/50 rule here!
Management of these extensions, much like the actual extensions, is limited. There isn't much you can do except see the available extensions and enable or disable them. This does come in handy in case you aren't seeing the extensions you think you have installed, or perhaps the list under your Share button is way longer than necessary.
Sometimes a software update can enable or disable an extension, or if an app is damaged somehow and Mac OS doesn't think it's installed anymore, the extension will disappear from this list. If you find your extension list isn't what you expected, here's where to start troubleshooting.