The others summed it up pretty well. Basically a reef is just a hard, shallow geologic structure, not necessarily built by coral or even anything living. In the past the major tropical reef builders have cyanobacteria, an extinct type of sponge, a type of calcifying sponge that was thought to be extinct until recently, an extinct order of corals, another extinct order of corals, an extinct type of bivalve, and most recently (since around the time of dinosaurs) it has been modern scleractinians. Notice that most of the former reef builders are now extinct and they almost all went directly from being the dominant organisms to being extinct overnight, geologically speaking.
Only about 10% of corals actually build reefs, so the end of reefs doesn't mean the end of corals or even the end of the species that build reefs.
To understand what's happening you need to think of coral reef formation as two competing processes. You have calcification from corals, algae, sponges, mollusks, etc. which is positive growth. You also have chemical, mechanical, and biological erosion, which is negative growth. When you add them together you get net growth. When net growth is positive you get reef building. When net growth is negative, you get no reef-building or the decline of existing reefs (aka a "give-up" state).
As the chemistry of the oceans changes, calcification becomes less energetically favorable so calcification (+) slows and chemical erosion (-) speeds up. More frequent bleaching also increases the energetic demands on the corals and slows calcification more, as well as reducing reproduction. Eventually, negative growth dominates and you enter a give-up state. At that point, individual corals will still grow, but when they die recruitment of new corals to take their place is low and overall the reef is eroding faster than the corals are calcifying. Reefs and the corals on them will persist for a while after that, but won't be spreading and as the corals die off they will be replaced by different organism (probably algae). If current trends continue, as soon as 50 years from now there could be no places on the planet that have suitable conditions for positive net growth.
