Where can I check all the installed Magento extensions in the admin panel?
There's a lot of confusion around terminology here. Magento, the programming system, has code Modules. These modules are often (incorrectly, although I've done it myself) called extensions. Modules allow you to add code to Magento's configuration based MVC system.
Magento also has a system called Magento Connect. Magento Connect allows you to install extensions. Magento Connect extensions are special pear packages. These packages can install a file anywhere in the Magento system. An extension may contain many Modules. It may also contain arbitrary files that are not part of a module.
There's no formal place to check which modules are installed in Magento. Informally, people use the Disable Module Output configuration section at
System -> Configuration -> Advanced -> Disable Modules Output
This section will create a list of each programatic module installed in the system
For Connect extensions, go to
System -> Magento Connect -> Magento Connect Manager
This is where you're supposed to install/remove/manage your connect extensions.
There's still the problem of manually installed extensions. Magento doesn't offer a formal marketplace. Many third party extension developers have created their own installer systems, or else have you install their files manually. There's no way to track these extensions other than diffing your code-base.
The original question asked how to 'check'. This could mean a few different things, such as what modules are installed, which are active, so on. Another thing to check would be version number, to see if there are updates out for your modules.
Many modules write their own management tools to save this hassle, but many do not- for those cases I find it easiest to check the database for module versions.
When an update script runs it adds a database entry into the core_resource table, this is how Magento knows not to run the update scripts over and over.
Simply check this table and you can find out all the versions of the modules that are installed in one spot. Doesn't solve the issue of finding out the current developer version- but, solves half the issue.
Other answers are of course correct, you can check in the admin panel under 'System'->'Configuration'->'Advanced' heading, click 'Advanced'.
This gives a list, just like if you were to look in the file structure under app/etc/modules at all of the xml files.
You can check your module settings using online tools. Especially it goes for shipping and payment modules (if you have any installed). Here's some more information on checking your estore configuration. In case it is incorrect, your customers may have trouble at the checkout.