Hi,
FaceTime can use just an iPhone Number on an iPhone (or iPad with active SIM card) or an Apple ID.
On a Mac it is normally an Apple ID.
However both can be linked and then you can answer calls sent to the phone number on your Mac can vice versa.
Both iMessages and FaceTime use servers that give some feedback as to whether they can see your Contacts are On Line.
(Similar to older messaging system like MSN Messenger or AIM but without the Buddy Lists).
When the app can't contact the server properly you get the aforementioned "error" alert.
It can occur if the server has a glitch or is Off line: See https://www.apple.com/support/systemstatus/
It can also occur if your Internet speed is not that stable.
In rare cases it can be that the Server end has interpreted you Login attempt as being "Too Many Too Soon" (even when you have not) and then it blocks you by device (Your iPhone may still login if your Mac is failing to do so).
In this case you have to phone Apple Care/Support and ask someone to look at the server info to find out if it has happened to then have it Reset.
This may mean asking to be passed to an higher level of responder that may have permission to look at the server info and have it reset for you.
Ralph