You can think of ArcCatalog as a user interface to a GIS server. ArcCatalog lets you view and, if you're an administrator, manage the set of services running on the server. ArcCatalog provides two distinct views of your GIS server, one for administrators and one for those with consumer privileges.
When you make a user connection to a GIS server, ArcCatalog simply displays the list of services available to you. You can use these services (for example, you can display a map hosted on the server in ArcMap), but you can't manage them in any way (for example, you cannot delete a service). When you connect to a GIS server as an administrator, you'll see some extra tools that allow you to manage the services as well.
ArcCatalog lets you administer the set of services running on your server and the set of machines that comprise the server. You can monitor how client applications consume individual services and whether there are enough resources to satisfy demand. At times, you may need to increase the amount of computer resources allocated to a particular service; other times you may need to add new computers to handle the load.
If you've just installed ArcGIS Server, there are a few things you need to do before you can start creating services and allowing client applications to access them. See the topic Getting started after install for a list of what you need to do.
A second way to administer the GIS server is through ArcGIS Server Manager, which is installed with the Web Applications component of ArcGIS Server. In addition to server administration capabilities, Manager contains an easy-to-use interface for creating Web mapping applications.
ArcGIS Server Manager can perform all of the administrative functions that are available in ArcCatalog, except for creating map and globe caches. This is because you must have access to the geoprocessing tools in ArcToolbox to create caches.