Monetary policy has two crucial roles. The first is to set the medium/long term inflation rate. Pretty well everyone understands this. The economy will not by itself settle down to an inflation rate of 2% or whatever - it needs monetary policy to set this rate and help achieve it.
The second is to ensure that aggregate demand matches aggregate supply. Now here there is sometimes confusion, even among the best economists.  The basic idea is that there is a ‘natural’ level of output determined by supply side factors, like how much people want to work, the degree of monopoly in the labour market, the state of technology etc.  There will be a real rate of interest associated with this level of output, which we can call the natural interest rate. On the other hand how much firms produce in the short run is largely determined by aggregate demand: firms tend to set prices, and do not ration demand. There is no reason why aggregate demand has to equal supply in the short run in a monetary economy. The difference between actual output and natural output is the output gap. If the output gap is not zero, problems will arise. For example with excess demand we get inflation, and with deficient demand we can have wasted resources and the misery of involuntary unemployment.
Aggregate demand depends on real interest rates. As monetary policy can influence real interest rates in the short run, then its job is to try and match aggregate supply and demand, by bringing the real interest rate as close as possible to the natural interest rate. 
These two roles for monetary policy map nicely into the two objectives macroeconomists typically ascribe to policy makers: minimising excess inflation and the output gap. With two goals there will also be conflicts, producing a trade-off between short run inflation stability and eliminating the output gap. Macroeconomics has extensively examined what to do when these conflicts arise.
A permanent non-zero output gap is not compatible with stable inflation in the long run. As a result, it is possible to reduce both roles to one single objective, the stabilisation of inflation, as long as that stabilisation is done ‘flexibly’. Hence the idea of a single, but flexible, inflation target. I now believe having only an inflation target, or making it 'primary', is an important mistake for two reasons. We can label each mistake as MPC and ECB for short.
The first (MPC) is due to persistent shocks to the relationship between the output gap and inflation (or equivalently to the Phillips curve). This sets up a potential conflict between the two goals. Although we know how to optimally deal with this conflict, the policy that results can appear inconsistent with inflation targeting, which puts a strain on an inflation targeting policy. The problem can be ‘solved’ by making inflation targeting even more ‘flexible’, but this in turn makes the policy less clear.
Of course macroeconomists have always acknowledged this possibility, but have thought that the impact of excess or deficient aggregate demand would always be strong enough for this not to matter in practice. However, as the recent IMF study I discuss here shows, either low inflation or credible inflation targets (or both) seem to have weakened the impact of the output gap on inflation, which makes the problem of persistent cost-push shocks more important. This has been the problem the MPC in the UK have been grappling with in recent years, and I believe the lack of a dual mandate has made their decisions less optimal. More generally, as Paul Krugman says here, thinking that stable low inflation must mean everything is OK could be very wrong.
The second (ECB) is that, in the wrong hands, the flexible inflation targeting regime can become a severely non-optimal policy that pays too little (or asymmetric) attention to the output gap, even in the absence of supply side shocks. In academic language, we could express this in terms of Rogoff’s conservative central banker (giving less weight to the output gap than the public does), but it also allows bad policy enacted by an incompetent central bank (that does not understand the importance of the output gap) or a malevolent central bank (that wants to achieve its own objectives that may not just involve hitting an inflation target).
There is a nice quote by Duisenberg from February 2003 contained in this paper by Jörg Bibow (page 35) that a comment from an earlier post pointed me to. In discussing what price stability meant, he said it “implies that, in practice, we are more inclined to act when inflation falls below 1% and we are also inclined to act when inflation threatens to exceed 2% in the medium term.”  Now Andrew Watt and others would argue that this is not a good reading of the ECB’s actual mandate, but it seems to me a good reading of what they actually do, and it is one that a single rather than a dual mandate helps them to get away with.
So what is the objection to a dual mandate? As it reflects how an academic thinks about monetary policy, it should not lead to suboptimal policy in the hands of an informed and benevolent policy maker. So the fear must be that it will misdirect an uninformed policy maker, and encourage non-benevolent behaviour. The exemplar here is the 1970s, but for reasons I discussed here, I do not think that period should be used as evidence against the dual mandate. I discuss here why I think the standard inflation bias story is also overrated in this respect.
Is there any evidence that the US with its dual mandate has done better compared to inflation targeters? There has been some discussion of that recently (e.g. here and here), although the data analysis is not very sophisticated.  Until we see good evidence that having a dual mandate worsens outcomes, then I believe the presumption must be that the dual mandate is better because it reflects the two goals of monetary policy.
My argument here concerns higher level objectives. It is not about how best to achieve those objectives, which is where I would locate questions about the wisdom of nominal GDP targets. It does not address the relative weight that the two objectives should have, or the extent that the objectives should be vague or concrete. My own view is that the benefits of a publicly announced inflation target are overwhelming - indeed so much so that I recently forgot how new this ‘innovation’ was for the Fed. How best to express the goal of matching aggregate demand with supply is more difficult, because of the uncertainties involved in measuring the output gap. However output gap uncertainty is not so great that we should ignore the concept, and so this uncertainty cannot justify a single inflation mandate. There are lots of things in life that are difficult to define, but which are still worth striving for.
 See, for example, Brad DeLong here. The reasons for this confusion are interesting, but I have speculated on this elsewhere and do not want to get distracted. Of course none of this implies that the natural level of output and its associated interest rate need be in any sense optimal or efficient, but that should be a different and separable question.
 There is nothing mysterious about the natural level of output. It is the output which pretty much every macroeconomist not investigating problems of aggregate demand analyse. It could be called the level of output that comes out of an RBC model, for example. It is often described as the level of output that would occur if prices were completely flexible, and here I do have a quibble, because at a zero lower bound and with inflation targets I cannot see how increasing the flexibility of prices will ensure that output reaches the natural level.
If monetary policy cannot do this, then fiscal policy can help reduce the output gap. We could describe this as fiscal policy raising the natural interest rate, but an equivalent and more intuitive description is that fiscal policy just raises aggregate demand.
There are plenty of caveats to this econ 101 account, such as the possibility that actual output might have an influence on longer term aggregate supply.
 If the implied asymmetric differentiation between actual and possible future here was a slip, I’m tempted to suggest it was a Freudian one.
 For example, MPC decisions since the recession have tended to define flexibility as ‘seeing through’ actual inflation and focusing on expected inflation two years out. Inflation targets then become a constraint when the impact of cost-push shocks persist for two years or more.