A managed user is most often a user that has been created within your admin console. As an admin or co-admin, you will have the ability to edit, delete, enforce security settings, and run activity reports on these users. Any files that these users upload into folders they own will count against your total storage allocation.
Our help pages on creating and editing Managed Users can be found here:
An External User is someone you have invited to collaborate in your Box account via a collaboration invitation. An External User is still a person who has a Box account, but the account was not created directly from within your own Box Admin Console. While you are still able to share and collaborate with External Users, your management capabilities over external user accounts are limited to the folders you have invited them into.
With certain paid accounts, both External Users and Managed Users contribute towards your overall purchased user seat count. Enterprise and Elite accounts have unlimited external collaboration, so only Managed users contribute to your seat count.
For further information, visit our Managing Users Best Practice.