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"Fiscal responsibility" (to be read as if surrounded by massive air quotes) generally refers to governments running smaller deficits, balanced budgets or budget surpluses even when it is economically counterproductive to do so. Advocates of "fiscal responsibility" propose policy change with the goal of achieving a lower deficit or larger surplus. Most commonly this involves spending cuts, but it can also include increasing taxes.
Austerity instills a fear in people, beginning with the lie that the federal budget is equivalent to a household budget and snowballing from there (debt ceilings, big government, "living within your means" etc.). Unless someone has taken some time to learn about macro-economics, there isn't a way that those lies can be dismantled. Advocates of austerity also push the narrative that government is all going broke, and that younger people are not expecting or cannot expect it to be there for them in the same way as in the past by the time they need it.
Detractors sometimes call it deficit fetishism or austerianism – after the wingnut pseudo-theory of economics called Austrian economics which it resembles, both in terms of lack of scientific methodology and in terms of conclusions – or, in the context of aid and development, the Washington Consensus – even though it's never really been applied to Washington for any very significant period of time, only by Washington to other countries, on the threat of loss of debt bailouts.
 Views of economists
Reinhart and Rogoff famously claimed to have shown that a public debt ratio of above 90% of annual GDP was dangerous, in a non-peer-reviewed issue of American Economic Review, and their study was triumphantly cited by George Osborne to give cover to his austerity programme in the UK. However, Reinhart and Rogoff's study was later shown to have been incorrect, due to a spreadsheet error, and confusing correlation with causation. Far from causing negative growth (recession/depression) high levels of debt were actually associated with above 2% average growth, similar to the growth of the other countries (though even then, causation could not be established). After this error was exposed, Reinhart and Rogoff used their fame to write opinion pieces in the media about austerity.
Many mainstream economists have written articles and signed letters denouncing austerity such as that practiced by the UK Conservative (nee Coalition) government, and noted that it is in fact austerity that is irresponsible, as it will slow growth and probably cause even employment growth to slow as well.
Post-Keynesian economics, and in particular Modern Monetary Theory, argues forcefully against the coherence of the simplistic right-wing notion of fiscal responsibility that is used to motivate austerity programmes, arguing that sovereign governments are not analogous to households, and the economically "right" level of government deficit or surplus entirely depends on the state of the economy at the time (e.g. the level of unemployment). Many of the relevant points made by MMT have also been made publicly by more mainstream economists as well.
 Political support
The principle of "fiscal responsibility", together with "sound money", is endorsed by many right-wing parties from various different countries, although this must be taken with a truckload of salt because it is actually quite unusual for any government to run a budget surplus.
Left-wing parties sometimes also endorse it, but this is mainly to attain "credibility" with floating voters, rather than because they actually believe in it. For example, in the UK, Gordon Brown pledged prior to the 1997 general election to follow Conservative spending plans for the first 3 years of a Labour government, and kept his promise, after which he was free to increase spending as he had wanted to all along. Ed Miliband tried the same electoral strategy, but his opponent David Cameron defeated him — in part by using the cunning gambit of promising to spend more than Miliband on the revered National Health Service. Clearly, Ed hadn't quite understood how the Brown principle worked - you don't go more Tory than the Tories on spending.
To a lesser or greater degree, depending on the individual member country and its level of influence within the EU, the European Union has advocated "fiscal responsibility". The fact that Greece has been forced into running continuous primary budget surpluses and has seen its GDP fall off a cliff — along with many other examples from the last few decades where creditors such as the IMF have forced a fiscal straitjacket on poor countries, with similarly dismal results — doesn't seem to have deterred advocates of so-called "fiscal responsibility" and its close cousin, "austerity". Nor does even Germany's relatively low levels of growth, or the low levels of growth of the Eurozone since its foundation and persistent high unemployment overall in the Eurozone.
Austerity theoretically means, in the words of David Cameron, "we're all in this together" (taking a hit to our wallets and/or public services for the long-term good of the country) — although in practice it means slashing government spending and disproportionately impacts the poor. How "austerity" can include (as it often does in practice) tax cuts for the top end of society – not merely at the completion of the supposed belt-tightening process but years before it has finished – has never been satisfactorily explained.
The Tea Party movement has long pushed for "fiscal responsibility". Quite how they intend to combine this with tax cuts is left unexplained. This culminated in the Congressional Tea Party Caucus, which advocates for budget surpluses.