Currency policy: The early to mid 1980s were a disaster, largely because Chilean banks were unsound and Chile pegged to a rising U.S. dollar. This was the biggest economic mistake of the Pinochet regime. It was a huge error which ruined the Chilean economy for years. It should be noted, however, that the move from 1000 percent inflation to 10 percent inflation is never an easy one.
Copper: Rising copper prices were a key part of Chilean economic success, and yes Allende had nationalized the copper mines. Pinochet did not re-privatize them. You can count this in his favor, but not in a free market way. Note that copper often counted for 50 percent or more of Chilean export earnings. You can talk about voucher privatization, but in fact the Chilean state needed a fiscal base to avoid the distortionary (but revenue-raising) policies that are otherwise so common in Latin America.
Agriculture and diversification: Chile moved from very high tariffs to virtual free trade. The Chilean economy diversified and became far less dependent on copper; this included some moves to hi-tech and light industry. The regime gets high marks on this score, as few modern nations have benefited more from free trade. Under Allende tariffs were commonly 100 percent or higher. Conception was the most rapidly growing Chilean city during much of the reform era, and this shows that the story is not just about copper.
Welfare: As in the New Zealand economic miracle, social welfare spending rose over the broader course of the Pinochet reforms. I take this to be for public choice and social control reasons, not benevolence. The distributional consequences of the Chilean reforms were generally inegalitarian, although there was the creation of a large and growing middle class.
Social security: The vaunted "privatization" was in fact done on top of an already-existing government-run system. Its free market nature is much overrated. It probably was a good idea at the time, given the stellar performance of the Chilean stock market (which it helped drive), but it has created long-run problems and it is no longer a system to be envied.
Privatization: Pinochet sold off steel, fertilizer, and an airline, among other companies. The procedures were corrupt, but nonetheless an improvement overall. There was no good reason why those companies should have been owned by the state.
Banks: Interest rates were freed up and financial repression was replaced by financial liberalization. Do note that the Chilean government took over the banking system in 1983 or so, to avoid collapse, but later returned it to private hands.
The Allende regime: It was ruining the Chilean economy. The recipe included deficits, foreign debt, high inflation, price controls, and devaluation, the usual stuff. The guy was brave, but as a leader we should not idolize him. He was an economic disaster.
Credibility: The Pinochet regime did restore the economic credibility of Chile. Investors came to expect pro-commercial policies. Although Pinochet himself was deeply corrupt (this was not known at the time), on net Chile extended its reputation as the least corrupt country in Latin America.
The 1990s: Much of the superior economic performance of Chile came after Pinochet left the stage (do look at the graph behind that link). But the roots of this growth spring from the Pinochet years. The moderate and left-wing successors left virtually all of his economic policies in place and of course they were democratic and eliminated the torture.
The bottom line: Many economic mistakes were made. The biggest gains came from agricultural diversification, general credibility, the rising demand for copper, and the move away from terrible economic policies. Chile had the best economic policies in South America from the late 1980s onward and it was and still is the envy of the continent. But Chilean history from this period is as much a "state-building" miracle as a "free market" miracle.
The Wikipedia entry on Chilean economic policy is reasonably good. Here are some other links. Out of the Ashes is the most comprehensive defense of Pinochet; it has useful material but overall it is not to be trusted. Google yields up many more critiques than defenses.
Pinochet the man behaved so badly, both during his term and after, as to be morally indefensible. From second hand accounts I have heard, it is also not clear how much the man himself was personally responsible for the good economic policies. Still many good policies happened. We need a closer look at the Chilean economic legacy, which is a complicated story and by no means wholly negative.
Addendum: It is worth asking which reforms could have succeeded in a democratic environment, but that would require a post all its own. Someday you will get it.