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Extensions are found in the extension catalog. By default they are installed to ~/.opera/widgets on Linux. An extension ends with .ocx and is a zip archive, inside of which is usually a copy of the license.

EXTENSIONS I USE
Privacy and security

• Ghostery is a behavioral advertising blocker. Unknown license.
• LastPass is an online password safe. Proprietary license.
• NotScripts is a JavaScript blocker. Open source.
• Redirect to HTTPS: Automatically redirects you to the secure version of web pages. Open source.

Usability

• AdBlock Plus and Opera AdBlock are ad blockers that improve on Opera's built in blocker. Unknown licenses.
• CleanPages improves the readability of web pages by removing clutter and enhancing the layout. Open source.
• Translator: Translates web pages or selected text using your choice of online machine translators. This is one extension I authorize for secure pages and private tabs. Unknown license.
• Up For Everyone adds a site availability check to Opera's "Network Problem" error screen when a requested URL is unavailable. It answers the question "is it just me, or is this site really down?" Open source.

Site-specific

• Facebook Adless removes advertising from Facebook pages. Unknown license.
• Facebook Blocker complements NotScripts (see above) for Facebook users. It blocks Facebook scripts from running while on third party sites, thus preventing Facebook from tracking your browsing behavior. When you're on Facebook itself, scripts run fine. Open source.
• SubDued hides comments on YouTube video pages, sparing you the ignorant drivel that passes for comments there. Open source.

Web development

• SeoQuake provides data useful for search engine optimization and web site promotion. Open source.

KEEPING UP TO DATE
Extensions are automatically updated.

TIPS AND TRICKS
If you're having trouble installing an extension, right click on the install button and download the extension's file to your computer. When downloaded, drag the file into Opera to install the extension. If it fails, the download was corrupted, presumably for the same reason the installation via normal means failed. Try again as needed.

I haven't tried it, but it is said that you can share extensions preferences between computers by using a synchronization service such as SpiderOak or DropBox.