Coase's most influential papers were published in 1937 and 1960. The first of these, entitled "The Nature of the Firm," gives a theory as to why companies exist that is still discussed in the field of organizational economics today. The second of these, "The Problem of Social Cost," shows how bargaining and assignment of property rights can alleviate the problem of externalities without the need for government intervention in the form of taxes or subsidies. (It is from this second paper that the fairly well-known Coase Theorem came about.)
In "The Nature of the Firm," Coase pondered why companies exist- after all, when Adam Smith wrote about self-interest and the invisible hand of the market, he referred to an individual bread maker, not to a multinational bread conglomerate. If markets work well to not only provide final goods and services to consumers but also to allocate resources used in production, it is more than a little perplexing to see a bunch of companies doing business using hired employees rather than a huge network of independent contractors explicitly selling their output. Coase'e explanation is that there are "transaction costs" associated with going to a market for everything a business needs- clearly, it would be inefficient to have to engage in a market transaction every time a manager needs an assistant to write an email, for example- and that these transaction costs are reduced when an employee is hired into a company simply because the company is essentially purchasing the right to decide on the fly what it wants to the empliyee to do (within reason, of course).
In "The Problem of Social Cost," Coase outlined what is now widely known as the Coase Theorem. The Coase Theorem states that “if trade in an externality is possible and there are no transaction costs, bargaining will lead to an efficient outcome regardless of the initial allocation of property rights.” In simpler terms, the Coase Theorem is most easily explained via an example, and, because Mr. Coase is fairly old, it's only appropriate to use a "get off my lawn" example. Let's say Mr. Coase has a part of his lawn where it's ambiguous whether the lawn belongs to Mr. Coase or to the public. Furthermore, let's suppose that some kids want to play on this part of the lawn and that the noise annoys Mr. Coase. If having peace and quiet is worth more to Mr. Coase than playing on the lawn is worth to the kids, it is efficient for the kids to be quiet, and vice versa. Coase's argument is that the efficient outcome will eventually be agreed upon (and potentially coupled with compensation from one party to the other) regardless of whether Mr. Coase ultimately has a right to quiet or whether the kids have a right to play on the lawn.
Even at the ripe old age of 101, Ronald Coase is still active in the economic world. Along with coauthor and fellow economist Ning Wang, Coase has a book coming out in 2012 entitled How China Became Capitalist. This work most notably argues that China's future growth potential will be limited unless it develops a flourishing market in ideas.