How repeaters work

This is a basic introduction to how standard radio repeaters work.
Repeaters are used to extend the range of a radio. They are usually put somewhere high up, so that there are no objects between the antenna of the repeater and the users it wants to serve. This allows the repeater to hear and transmit much further than a person on the ground. Using a repeater you could expect as much as 10 times the range or more compared to using handheld to handheld.

You may hear about offsets when repeaters are mentioned, this is the difference between the transmitting frequency and the receiving frequency. (called TX and RX for short)
The repeater listens for transmissions on its RX frequency (or input), and then when it hears a transmission on that frequency it transmits it back out on the TX frequency (output). The frequencies must be a certain distance away from each other, otherwise when the repeater starts transmitting, it will desensitize the receiver and will no longer be able to hear the user talking.
Radios will be programmed to listen on the output of the repeater, and to transmit on the input. This means they will always hear any transmission that the repeater makes, and the repeater will hear them whenever they transmit. The radio hears other users via the repeater, even if the other user is right next to them. If they are out of range of the repeater they will not be able to hear anything even if another user is transmitting right next to them, because they will be listening for the repeater to transmit that message.

Generally when scanning you will hear two types of repeaters. Amateur repeaters and business radio repeaters. Amateur repeaters are open to access by anyone with an amateur radio license, which can be got from ofcom by sitting a simple test to show you have an interest and a basic knowledge of radio. Business repeaters are closed to the public, and are only for use by employees of the business or people they allow to use the repeater. You will hear businesses like taxis, shopwatch schemes, local authorities, parking wardens who will usually have their own repeaters.
Amateur repeaters have fixed offsets in each band, for example the 2m band, which goes from 144-146MHZ, has an offset of +600KHz. This means that for all repeaters in that band will have their input frequency exactly 600KHz above their transmit frequency. For example: If you hear a repeater transmitting on 145MHz, to transmit into the repeater and have the message repeated back out, you would have to transmit exactly 600KHz higher, which is 0.6MHz. So you would have to transmit on 145.600MHz. (Which you would of course only do if you had a license to do so!)
Business repeaters will have varying offsets, but they are often the same in certain ranges set by Ofcom. The frequencies of repeaters can be found on the Ofcom WTR (Wireless Telegraphy Register).

Repeaters will also usually have an extra requirement to satisfy to be able to use them. This is a CTCSS (Continuous Tone Coded Squelch System) or DCS (Digitally Coded Squelch). People often refer to these as a tone, code or PL, which you can assume means CTCSS or DCS.
To put it simply, these are sub-audible noises that are added to your transmission and then picked up in receiving radios/the repeater and decoded. If you transmit with the wrong CTCSS/DCS, or without one, the repeater will completely ignore your transmission. Once you have the correct one set, the repeater will hear you and transmit your message out on the TX frequency, often with the same tone/code.
Using the same CTCSS or DCS on the repeaters transmission is useful because it stops the receiving radios from picking up and other users who may be on the frequency (unless the also have the same tone set), or interference caused by electronic devices being close to the radio.
Scanners also sometimes have CTCSS and DSC which is used to filter out different groups, or interference on a channel, and only hear the group you want to hear who will be using the same code.

Digital repeaters will not use CTCSS or DCS, but they will use other software parameters that are set in the programming software. In DMR this would be the colour code. DMR repeaters work in a very similar way to analogue repeaters, but they listen on the input frequency and then receive and demodulate the data. Then they will modulate the data again and transmit it on the output frequency, often adding data on one timeslot, such as "slot empty" messages when only one timeslot is in use.
Because users transmit on the same input frequency, but on different timeslots, the repeater can repeat two voice messages at the same time by simply listening to both timeslots on the frequency. The output of the repeater would repeat both timeslots and receiving radios can listen to either of the timeslots depending on which group is set to use which timeslot. So if one group of users is programmed to use slot1, another group of users who have their radios programmed to use slot 2 will not hear transmissions by the first group because their radios are listening on the wrong timeslot.