I just received this question from a reader who is doing a research project on classifying contemporary schools of economics. The problem is that the question is not as easy to answer as one may think.
Clearly the contemporary Austrian school of economics is not part of the mainsream of economics. But many of the main themes in Austrian writers are consistent with arguments presented by classical and neoclassical writers from Adam Smith to J B Say to Milton Friedman. Austrians are part of the mainline of economic thinking, but not currently part of the mainstream.
This produces a weird intellectual context for me to answer the original question. Part of the reason the Austrians are not part of the mainstream is because in the 20th century they resisted the unholly alliance of statism and scientism. Ironically, the more difficult argument to win is the one against scientism --- statism while hard is not nearly as hard to defeat. But the "science wars" are very difficult to win unless you ape the methods of the physical sciences. Do a thought experiment for minute and consider the following developments in economics over the past 50 years: law and economics; public choice; rational choice sociology; new economic history; mathematical economics; and experimental economics. In each of these cases, the field that was perceived to be the more "scientific" influenced research, not the other way around. In most of the cases, economics was the more scientific field, but in the cases of mathematical economics and experimental economics, the more rigorous field invaded the less.
But what if while economics is definitely a science, it is a human science; a philosohical science, that is unique and distinct from the natural sciences. To put it another way, what if we agree that the way to advance the natural sciences was to purge human explanations where they didn't belong (overcoming anthropromorphic explanation), but when applied to the human sciences this intellectual thrust would purge the very thing we hope to study?! In other words, while purging anthropromorphism in the natural sciences resulted in progress, in the human sciences all that results is mechanomorphism. Zesus didn't cause lightening bolts due to his anger anymore than economic systems operate like clocks or bathtubs or even linear accelerators. The human element cannot be purged from economics and for us to make intellectual program.
Fighting scientism is much harder than fighting statism, and thus even though the Austrian school originally fell out of fashion due to the rise of anti-market arguments, when market-oriented arguments came back in fashion the Austrian school remained for the most part on the outside looking in with respect to the economics profession.
But ironically many leading mainstream thinkers come back to Austrian points at the back end of their careers as they start to realize that standard models miss out on dynamics and creativity. To use the langugae of Nelson and Winter, they realize that the appreciative theory of the market found in Mises/Hayek/Kirzner outdistances the formal theory of the market found in Arrow/Hahn/Debreu.
So what is my answer to the question? Austrian economics starts from within a heterodox problems situation (heterogenous agents, uncertainty, ignorance, etc.), and they interact within complicated environments (differing tastes, heterogeneous products, market power, etc.), and thus the Austrian answer to how markets work differs considerably from the standard presentation in textbooks, and the non-standard presentation in heterodox writers. Instead, the focus is on how the institutions of property, contract and consent engender market processes of adjustment and adaptation that produce social cooperation under the division of labor.
In other words, the Austrian school is heterodox sociologically, mainline intellectually (tracing back to Adam Smith and David Hume), and mainstream historically (a branch of neoclassicism). But in the contemporary world, the mainstream label makes the least sense, the mainline label is little understood let alone appreciated, and the heterodox label serves a useful sociological function. Though the Austrian school doesn't really fit anywhere easily.