Bunker fuel is technically any fuel that is used by a ship, and therefore stored in a port's bunker tanks. However, 'bunker fuel' is usually used to refer to the poorest of combustible petroleum, only one step up from asphalt (bitumen). Depending on context it may also be referred to as heavy oil, resid, blended fuel oil, or furnace oil; however, in some cases these terms might also be applied to other types of petroleum products. The most appropriate name is probably #6 oil or Bunker C (#6 oil mixed with some #2 oil), although these labels are rarely used outside of the fuel and shipping industries. The press has firmly landed on the term 'bunker fuel', and thus so have I.
Bunker fuel is not actually an inefficient fuel, as far as energy produced goes. It has a slightly higher combustion efficiency than does natural gas, although it has its fair share of downsides: in order to be used effectively it needs to be preheated to reduce viscosity, have additives to prevent corrosion of the fuel tanks pipes, be stirred frequently while in storage to keep the additives from separating, and it produces quite a lot of soot when burned.
Bunker fuel has edged into the news lately for environmental reasons. There are two generally cited reasons that bunker fuel is "bad". The first is that it burns somewhat dirtily. A large part of this is simply the aforementioned soot, but bunker fuel also contains heavy metals and slightly elevated sulfur oxide levels. Much of the emissions problem is due to the fact that the ships that burn this fuel are comparatively unregulated (power stations, cars, and even kerosene heaters, by contrast, are quite heavily regulated). Much of the laws governing emissions only cover, or are only enforced, when the ships are in port. The UN's International Maritime Organization and other environmental organizations are working to change this, so hopefully the shipping industry will soon become more environmentally friendly.
The second environmental problem with bunker fuel is that when the oil spills, it can remain in the environment for quite long periods of time. NOAA reports that bunker oil, being quite viscous, is comparatively easy to clean up early in a spill. However, it goes on to explain that #6 oil is likely to evaporate very slowly, and to form tarballs that can drift for hundreds of miles. It is also dense enough that in some cases it can actually sink, polluting all levels of the oceanic ecosystem. Once it washes up on shore it is likely to remain on the surface (while thinner oils may soak in), causing even more havoc for local wildlife. If it is not cleaned up quickly, it will 'cure' into a sticky, enduring mass. On the plus side, #6 oil is less toxic than many higher grade oils, and thus its effects are less likely to travel through the food chain. A lot of the bad press bunker fuel has gotten is simply due to the fact that it is what is spilling; if ships were spilling a different fuel, that fuel would be villainized instead. Diesel fuel, also a common fuel in the shipping industry, has proven to be much more toxic to plants than is bunker fuel. The best solution, of course, is to avoid spills altogether.
But why, you may be wondering, do ships use bunker fuel at all? Burning tar-like sludge seems odd, at best. Part of the reason is that bunker fuel isn't generally in great demand for other applications (keeping the price low), although industrial boilers do use it. There is, however, another reason. Crude oil is refined through the process of fractional distillation; in this process small hydrocarbons are 'boiled' off before larger hydrocarbons. The smaller hydrocarbons burn more easily, and faster. The larger ones burn slower. Cars run really well with a comparatively quick-burning fuel, but boats are different. When a motor is powering a propeller, there is a limit to the speed that it is practical for it to turn, and thus slow speed engines are adequate (and simpler, as they don't need as complex a gearbox to step down the engine speed) for nautical use. Thus ships generally prefer to use slower-burning, large-molecule fuels such as diesel and bunker fuel.