A radio repeater or just repeater is a device typically mounted to a large antenna tower used to amplify signals. Imagine that one radio operator wishes to talk to talk to another operator 20 miles away, but they are too far away to talk "simplex", or direct radio-to-radio. Therefore, the operators decide to use a repeater instead. There are two frequencies on a repeater, an uplink and a downlink. Let's imagine in this example that the uplink is 146.82 and the downlink is 146.22. User "A" transmits his message on the uplink frequency which is recieved by the repeater. User A knows that the repeater recieves his signal because when he finishes his transmission, he hears a short static burst. This static burst, sometimes accompanied by a beep, is called the repeater squelch tail. It means the repeater has properly recieved the signal. If User A did not hear the courtesy tone, he knows that either his radio settings are configured wrong or he is out of range of the repeater. However, User A did hear the response. As soon as the repeater begins to recieve the transmission of User A on the uplink frequency, it re-transmits on the downlink frequency. User B is monitoring this downlink frequency and after User A finishes his transmission, User B begins his transmission. The repeater antennas in this example are mounted on a high antenna tower and act as a sort of middleman between User A and B. The operators aren't able to hear each other's transmission but the repeater can. Repeaters can be permanent installations or deployable as go-kits at high points in emergencies, acting as a stable communications hub.
There are multiple good sources to find repeaters in your area. The most popular are repeaterbook.com and radioreference.com; I personally prefer RepeaterBook but that is personal preference and both services have comprehensive databases of pretty much all public-access radio repeaters in North America.