Google's Chrome OS, like Android, Apple's IOS, and Windows 8, has its own application store at: https://chrome.google.com/webstore/. Note that many apps in the store are little more than links to an external web page. I have not listed those in my list below, as they offer not much more than the convenience of having an icon with a link.
There are still relatively few selections here compared to the mature IOS and Android app stores, but I'm hopeful that more will be coming along as Chromebooks become more accepted.
The following is my running list of Google Chrome/Chromebook apps that I am using (not a complete list of apps available). This list is specifically limited to apps which can function offline. Online apps which require an internet connection (e.g. Pandora or YouTube) can be accessed just as easily from a bookmarked web page. Aside from the usual browsing and email use, my use of Chrome and a Chromebook is to access other machines (Windows, Linux) and do some light editing of text notes, web pages, and images. On trips, I use the Chromebook for offline entertainment (music, video, ebooks), and backup of photos and video from my cameras.
- Offline GMail - allows for browsing and editing email even when disconnected from the internet. When you're connected, it will synchronize drafts and send out all email you have "sent" while offline.
- Google Keep - a simple app for taking text notes, clipping web page links, or making links. This works in offline mode and synchronizes when connected to the internet. It's a nice, simple alternative to Evernote and similar note taking apps.
- Chrome Remote Desktop - Google's remote desktop server/client application for systems located on a local LAN or across the internet. The server does not currently work on a Chromebook, but works fine on a PC and Mac (not tried personally). The server machine does not have to have the Chrome browser open. The limitations are that you must be logged into your Google account on the server, or someone at the server must manually give you an access code. Also, on the client, only one server connection is allowed at a time.
- Caret - a full-featured text editing program with support for many specialized file types (e.g. HTML). This app works offline.
- Secure Shell - Terminal emulator and SSH client useful for server maintenance, whether located on the internet or internal LAN.
- Kindle Cloud Reader - a reader for Amazon's Kindle books. Books in your Kindle account can be downloaded for offline reading.