WA9ZZZ's answer missed the issue of control of the repeater. The repeater must be under control of one or more control operators, or under automatic control, at all times when the repeater is on. There are three types of control:
- Primary control, where a control operator is physically near the equipment and monitoring the repeater, ready to intervene as necessary, the whole time it's on.
- Remote control, where the control operator(s) is/are monitoring the repeater remotely and also controlling it remotely through a control link, typically a telephone line or a radio link.
- Automatic control, where devices and procedures are used to ensure that the repeater is operating in compliance with FCC rules. For instance, a repeater controller can time out and stop transmitting if someone has been talking continuously through the repeater for too long.
Most repeaters in the US have a repeater controller, which is a device that provides automatic control. Typically the repeater controller also allows some remote control to turn functions of the repeater on and off; for instance, a sequence of DTMF tones can be sent to turn the repeater off if someone is transmitting profanity through the repeater.
If remote control and not automatic control is used (no repeater controller), then the control link has to be usable 100% of the time the repeater is running. In my opinion this rule would preclude DTMF tones on the repeater input frequency being used for control, because a malicious operator with a bigger signal could block out the control operator.
To sum up, if you're not planning on babysitting your repeater locally or remotely, then you'll need a repeater controller so your repeater will legally be under automatic control.
I found much of this information on the ARRL's Auxiliary Station FAQ page.
Also, the repeater must identify itself periodically, like any other station. Repeater controllers can typically be programmed to do this, or I suppose a control operator could identify for the repeater.
Someone asked if a repeater needs its own call sign, because the owner of the call sign, the licensee, might be operating on a different frequency at the same time. As I understand, a repeater doesn't need its own call sign. Furthermore, an operator can be on several frequencies at once, from several different physical locations, as long as he or she is exercising control over all the stations. (Hat tip to @hobbs-KC2G.) Many repeaters identify using Morse code, and the nearly-universal practice is to put the suffix "/R" on the end of the call sign, which identifies that the call sign is for a repeater. If the identification is by voice, either by a recording or a live control operator, then I'd think the identification would be something like "this is W1AW, repeater". So the "/R" or "repeater" suffix on the call sign clearly identifies the call sign as being for a repeater, so there should be no need for a separate call sign.
In the United States, the only way that I know of to get a new separate call sign for a repeater is to use a club call sign. This process apparently was frequently abused in the past, because the rules have been tightened down considerably. First, there must be a club with at least four members, and the club must have a document of organization (typically a charter) and several officers, and it must have regular meetings. The club must designate a person to serve as trustee for the club call sign in a document signed by club officers. A person can serve as trustee for only one club call sign. A club can have multiple club call signs (and therefore multiple trustees), but only one call sign can be a "vanity" call sign. More information about club call signs can be found on the ARRL's Club Call Signs page.