An Assign contact is one in which specific agents are assigned to distinct contacts. It is applicable only for Contact based IVRs. So, whenever, a customer listed in the contact based IVR calls, the IVR menu (press 1 for sales, press 2 for support, etc) is skipped entirely and the call is directly forwarded to the assigned agent. For example, if he Account manager for caller A is Mr. X then, whenever customer A calls, the call is directly forwarded to Mr. X. 

Note: At present, this feature is available and accessible only through APIs and not via the web panel.