A control operator is a licensed amateur designated by the station licensee to be responsible for the station's transmissions and for following the rules. A station may never transmit without one. If you own and run your own station you are both licensee and control operator; if someone else is at the controls, both of you answer for what the station does.
No control operator, no transmitting. Both of the first two boxes answer for what the station does.
Privileges follow the operator
The privileges in force at any moment are those of the control operator's license class, not of the owner or the radio. An Extra who lets a Technician operate gets Technician privileges, and the Technician cannot operate in Extra-only segments (except in an emergency).
The owner's license, a trustee's license, or who else is in the room changes nothing.
The same logic applies to relays. A repeater that transmits on 10 m is transmitting in an HF band, so its control operator needs HF privileges: a General or higher. A Technician may still use it from the 2 m side. For a station working through a satellite, any amateur whose privileges include the uplink frequency may be its control operator.
Where the control point is
The control point is the place where the control operator function is performed.
The control point is wherever the control operator function is performed.
Control
Where is the operator?
Notes
Local
At the station
The usual case at home
Remote
Elsewhere, linked to the station
Any station may be remotely controlled, for example over the internet
Automatic
No operator at a control point
Repeaters; some limited digital stations
Sending CW or an ID by computer, or operating over the internet, is not automatic control. Automatic means no operator at a control point.
Responsibility in practice
Remote stations need a way to shut down. If the control link fails, the station must stop transmitting within 3 minutes. Whose rules apply depends on the station: running a US station from abroad needs only your US license, while running a foreign station from the US means that country's rules.
Automatic stations have limits on what they may carry. Third-party traffic under automatic control is allowed only with data or RTTY, never voice or CW. Automatically controlled digital stations may talk to each other only in the places the rules allow.
Repeaters and forwarding do not move the blame. If something improper is relayed, the control operator of the originating station is responsible.
Guests can speak on your station under your control operator's supervision. A person without a license may take part only while a licensed control operator is at the controls. See Part 97 in plain words for third-party rules.
Club stations are licensed to a trustee and at least four members, with a control operator for each session. See Clubs, Elmers and community.