The original purpose of FM translators was essentially the same as for TV translators. However, FM translators in the United States were much more restricted. Commercial FM stations can't own a translator outside their service area. Translators of an FM station can be independently established, and the owner of the translator can seek financial support from the originating station, but the translator can't be constructed or operated by the originating station. In most cases, translators have to be fed with an off-the-air signal (though that can be obtained through a CATV connection!). There is a big exception: Non-commercial educational (NCE) FM stations are not subject to these restrictions. Certain religious operators have exploited this loophole (in my opinion, abusing the purpose of translators to start with) to extend their reach nationwide, feeding many of their translators via satellite. As part of the FCC's AM "revitalization" initiative, AM stations were allowed to establish FM translators as long as the coverage of the FM translator did not exceed that of the AM station. Daytime-only AM stations, of which there are now very few, were allowed to continue broadcasting on their FM translators at night. The idea was to get AM stations a foothold on the FM dial where a majority of radio listening occurs and thereby give them a chance to compete. These are referred to as cross-service translators.

Another loophole involves programming originally on an HD (digital) subchannel. Those, too, can be relayed by a translator, giving an HD signal an analog FM presence. This is desirable because HD-capable receivers are not common, except for vehicles.

As for boosters: If you hear one, you probably won't know it. They are on-channel, intended to provide an infill signal for areas where FM reception is blocked by terrain, tall buildings, etc. (I have never heard a legal ID for a booster.) Good examples are the San Francisco Bay Area in California, just east of me in Oakland. On the other side of the 1500-foot hills separating Berkeley, Oakland, and San Leandro from the Diablo Valley (sometimes also called the Tri-Valley, the San Ramon Valley, etc. - essentially the Interstate 680 and Highway 24 freeway corridors and surrounding communities), there are several boosters relaying San Francisco FM stations. Most are on Mount Diablo or on Pleasanton Ridge. While they work, some are reportedly not well maintained. When I've been over in San Ramon and vicinity, I have noticed some multipath interference among the least well-maintained boosters. KITS was particularly awful - but I might have been receiving a badly ghosted over-the-air signal from their main San Bruno Mountain site. It's impossible to tell.