Admin Server is the central domain configuration interface which is used to create, delete and configure the resources of a domain like managed server, machine, cluster, data source, work managers etc.
Managed server is a server in a domain that is not the Administration server. It contacts the administration server for configuration information. It runs a business application in a production environment. It is independent of all other Managed servers in a domain (unless they are not in a cluster). You can have many managed servers in a domain. Individual managed servers are typically added for capacity and application isolation.
The Administration Server stores the master copy of the domain configuration, including the configuration for all Managed Servers in the domain. Each Managed Server stores a local copy of the domain configuration file. When a Managed Server starts, it connects to the Administration Server to synchronize the configuration. When the configuration is changed, the Administration Server sends the changed configuration to the Managed Servers.
There is no difference between Admin and Managed server except admin interface application by default deployed on admin server which is used to configure the domain resources. Apart from that, an admin server is same as a managed server, you can deploy and create other relative resources on admin server also as you do on managed server.
MSI is nothing but Managed Server Independence. By default, Managed Servers can function independently of the Administration Server. A Managed Server instance can start in MSI mode if the Administration Server is unavailable. Configure MSI mode from the Administration Console. To start a Managed Server in MSI mode, perform the following: Ensure that the Managed Server’s root directory contains the config subdirectory. If the Config subdirectory does not exist, copy it from the Administration Server’s root directory. Start the Managed Server at the command line or by using a script. Environment > Servers > Server_Name > Tuning > Advanced > Managed Server Independence Enabled check box.