A repeater is an automatic station that receives your signal on one frequency and retransmits it, at the same time, on another. Placed on a hill or tall building, it lets low-power radios reach far beyond what they could manage directly.
Two stations too far apart for a direct contact can both reach the repeater on its hill.
Receiver, controller and transmitter in a loop, sharing one antenna through the duplexer.
Receiver. Tuned to the input frequency. It needs to be sensitive, because it has to hear a handheld's small signal.
Transmitter. Sends on the output frequency, usually at much higher power than a handheld.
Controller. The brain. It detects a valid signal (and the access tone, if required), keys the transmitter, passes the audio through, enforces a timeout timer, sends the station ID and courtesy tone automatically, and may link to other repeaters or accept remote commands.
Duplexer. Lets one antenna be shared; see below.
Power. A dedicated supply, often with battery backup so the repeater works during a power cut.
Cavity filters let one antenna share transmit and receive.
The transmitter is far stronger than the signal the receiver is trying to hear, and the two frequencies sit close together, typically 600 kHz apart on 2 m and 5 MHz on 70 cm. Without filtering, the transmitter would drown out the receiver (desensitising it). A duplexer uses high-Q cavity filters, tuned to each frequency, so one antenna sends and receives. The alternative is two antennas separated vertically so little power couples between them.
In practice
Coverage is lopsided. The repeater may reach you while your handheld cannot reach it. Good sites and sensitive receivers matter as much as repeater power.
A repeater is a shared resource. Frequencies are coordinated to avoid interference; the people behind it pay for the site, power and upkeep. See Repeater directories.
Linking. Controllers can tie repeaters together by radio or the internet; see EchoLink, IRLP and AllStar.
Digital repeaters (DMR, D-STAR, Fusion) add the equipment needed to decode and regenerate the digital signal.