A repeater is a station, usually on a tall building or hill, that listens on one frequency and retransmits what it hears on another, at higher power and from a better antenna. A handheld that could barely reach across town can then talk across a whole region.
Two stations too far apart for a direct contact can both reach the repeater on its hill.
It must receive and transmit at the same time, so the two frequencies must differ. If it used one, its own output would drown out the weak signal it is trying to hear. The hardware is covered in Repeaters and controllers.
Setting your radio
You need four things, all listed in a repeater directory: the repeater's output frequency (where you listen), the offset direction and size, an access tone, and the mode. The radio works out the transmit frequency.
Example values only. Your radio's menu may call the tone field TONE, T-SQL or PL; the repeater directory gives the right numbers.
Offset direction follows the part of the band. On 2 m in the US, repeaters whose output is below about 147 MHz typically use a minus offset (you transmit lower than you listen); those from 147 MHz up typically use plus. Don't rely on memory; use the directory.
The reverse key swaps your receive and transmit frequencies, so you can listen to the repeater's input. If you hear the other station strongly there, you are close enough to talk direct and leave the repeater free.
Tones
Many repeaters ignore signals that don't carry a quiet CTCSS tone below the voice band. This stops them being triggered by distant stations on the same frequency. It is an access gate, not security. Some repeaters use digital-coded squelch (DCS) instead.
CTCSS: a tone below the voice band, sent with your voice, opens the repeater's squelch.
No tone, or the wrong one, and the repeater does not answer, though you can still hear it. A common beginner mistake. See Squelch, CTCSS and DCS.
Using one
Listen first. If a conversation is under way, wait.
Just give your call sign to let people know you are around: "W1ABC listening". There is no need for "CQ" on a repeater.
Pause between transmissions so others can break in, and keep each over short. Repeaters have a timer that shuts them off if you hold the key too long.
Identify with your call sign at least every 10 minutes and at the end of a conversation. The repeater's own identification does not count as yours.
Use the lowest power that works. If the repeater already hears you clearly, extra power adds nothing.
Don't "kerchunk": keying up without speaking to test it. Say your call sign instead.