A repeater receives a signal from a user’s radio and simultaneously transmits (repeats) that same signal on another nearby frequency through the same antenna. The difference between the input and output frequency is the offset (600 KHz on 2M and 5MHz on 70cm).
The offset is positive at one end of the band and negative at the other. All modern VHF/UHF radios have those offsets automatically incorporated into the factory programming for duplex operation, although they can be modified if necessary.
The frequency label of a repeater is its output frequency, which then becomes the receive frequency that you program into your radio for that repeater. When you transmit to that repeater in duplex mode, your radio automatically shifts its frequency by the offset to match the repeater input frequency.
Within any specific ham band there are band plans agreed upon by amateur radio groups that determine what parts of the band are to be used for repeaters, simplex, or other special modes.
Those plans are described in the ARRL Repeater Directory, which is a must-have for any ham using repeaters. It’s available at HRO and many other sources. The several chapters at the front of that book have a wealth of information on repeater operation. Especially see the chapter on repeater lingo/hints.
Sub-audible CTCSS (Continuous Tone-Coded Squelch System) tones, aka PL (Private Line) tones, are tones at audio frequencies below the speech range that are transmitted along with the user’s voice by the radio accessing the repeater. These are needed in order for some repeaters to respond. This is a means of preventing interference between repeaters in different locations on the same channel.
When programming a radio for a repeater frequency and its offset, the CTCSS tone of the specified frequency must be selected if that repeater requires it.