I dont know what software you have. If you dont have ArcGIS for Server, you cannot make your own arcgis server (server connection). Its not a website. Its server (machine) where are running some map services.
Example: Iam using arcgis server of State Administration of Land Surveying and Cadastre (ags.cuzk.cz/arcgis/rest/services) to upload detailed Imagery map of Czech republic or Hillshade into my ArcMap (ArcGIS online). There are lots of servers like this one or by ESRi etc. Type in google arcgis/rest/services
To create your own server you need 1) server (machine or cloud) and 2) instaled ArcGIS for Server on it.