A fuel surcharge is simply a mandatory fee charged by an airline, typically per-segment (per flight) and broken out of the actual fare for auto-pricing. For example, an airline may offer a fare from Seattle to Paris that is valid with multiple routings, e.g., SEA-LAX-LHR-CDG. The fare is published end-to-end (SEA-CDG), but the airline may impose a mandatory fee for each flight that is based on the flight distance or airports. E.g., the SEA-CDG fare might be $450, and they might impose a fuel surcharge of $500 for the LAX-LHR flight. Your actual cost (and the true fare) would then be $950 plus government taxes and fees. The airlines originally added fuel surcharges when the cost of fuel was going up rapidly and they didn't want to publish updated fares every day. These days, fuel is a fraction of what it cost then, but many airlines retain their high fuel surcharges.
Some airlines lump fuel surcharges in with taxes and government-imposed fees, but they are purely an airline-imposed charge that goes to the airline. One reason airlines love fuel surcharges is that they are technically not part of the fare. Many airlines have contracts with major companies and offer discounts which typically are calculated on the fare, so fuel surcharges are exempt. Also, airlines can pretend that the surcharge is a tax and thus you might blame "the government" for the high cost of the ticket.