The brief way to describe this is to say that Ronald Reagan wanted there to be lower taxes. He wanted the government to make fewer laws regulating what companies could do. Finally, he wanted to cut government funding to various types of social programs.
Pres. Reagan wanted to do these things because he was a conservatives. Conservatives, in general, want there to be less government. They especially think that the government should not regulate businesses -- they do not like minimum wage laws, for example. Conservatives also thing that social programs like welfare should be reduced so that people would have to find work instead of being able to live off government help.
To quote the great communicator himself, "Government is not the solution to our problem, government is the problem." The Republicanism espoused by Reagan featured a drastic deregulated approach to government. It is during the time of Reagan where government was seen as a bad thing. It was the source of intrusion and hampered the creative and innovative spirit to American commerce and its ability to assert its greatness. Taxes and regulation were representative of this, and Reagan's approach called for minimizing both as much as possible. Social programs that called for government intervention were strictly reduced and a more innovative spirit was called upon by the government. The notion of "individuals rising from their bootstraps" in order to demonstrate a sense of "national character" was invoked in order to justify the lack of government presence in the realm of social programs. In the end, this particular philosophy made certain people extremely wealthy, and had the opposite effect on others.
If one looks more closely at the Tax Reform Act of 1986, you see what has been previously stated in slightly more detail. Reagan set up a deal that would appeal to both liberal democrats as well as conservatives in an attempt to further his and his party's idea of trickle-down economics. They felt that by lowering the overall tax rate on the highest income earners but at the same time closing loopholes, they could reduce the tax burden on lower-income citizens while in some ways doing the same for wealthier citizens. This appealed to both sides of Congress and was something of an experiment in what was referred to at the time as "supply side economics," basically the assumption that if there was more money out there to be spent (rather than being taken in through tax revenue) that people would spend more, further increase not only the supply but the speed at which money moved through the economy helping out everyone along the way including the government by generating more revenue in the long run.