Chrome Web Store apps and Custom apps are apps specifically designed for Chromebooks. While Chrome Web Store is the Chromebook equivalent of Google Play Store, Custom Apps are enterprise apps exclusive for Chromebooks.
Web Store apps are the apps available for public download on the Chrome Web Store. These apps can be installed on Chromebook and on Chrome browsers. Chrome apps contain both productivity as well as education apps which can be used in schools and/or enterprises. To distribute and install a Web Store app on Chromebooks, refer to this.
You can install Android apps available for public download on Google Play Store on compatible Chromebooks. Play Store apps contain both productivity as well as education apps which can be used in schools and/or enterprises. To distribute and install a Play Store app on Chromebooks, refer to this.
In certain circumstances, organizations might require apps to cater to specific needs which might not be available on the Chrome Web Store. The answer to this is Custom Apps - Apps specifically built for a particular need or set of needs. These apps are usually created by third-party developers and available for download/use only on request as it is a private enterprise app. You can know more about creating a custom app here and publishing a custom app here. To distribute and install a custom app on Chromebooks, refer to this.