For example, we can simplify "a person who is invited by somebody" as "invitee". And does a short form exist for "a person who invites another person"?
Inviter is someone that invites.
Generally, the thing that instigates the action will have an -er ending, and the thing that is in receipt will have -ee ending.
Inviter - someone who invites
Invitee - someone who is invited
Employer - someone who employs
Employee - someone who is employed
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@EdwinAshworth, this is true. And has been suggested 'host' or some other word might be more natural, depending on context. – Graham Nicol Oct 8 '15 at 9:05
It could be called a "host", for one, in case he invites people to a party which he is holding.
I think "Host" is the closest answer, although in some cases it could be "Sponsor".
However, I don't think "inviter" is the answer at all (according to dictionary.com) in reality there is no straightforward word in English for that, I guess it has skipped usage all these years.