Efficient markets
An efficient market is one in which securities prices reflect all available information. This means that every security traded in the market is correctly valued given the available information.
There are a number of different definitions of what constitutes an efficient market depending on the what information is deemed to be available.
The weakest form of efficient markets is that securities prices reflect all information contained in historical prices. This is the easiest to prove, by showing that share prices follow a random walk.
The strongest form of efficient markets is that prices incorporate all information that any investor can acquire. This seems unlikely given that insider traders can undoubtedly make money fairly consistently.
The semi-strong form of efficient markets is that securities prices incorporate all publicly available information. Given how difficult it is to find groups of "smart" investors who consistently outperform the market, this seems likely. There do seem to be some investors with very impressive records. The semi-strong efficient markets hypothesis is probably very close to being true, but not always true.
The most glaring exceptions to efficient markets seem to occur during investment bubbles and collapses when prices reach levels that can not be explained by reasonable valuation methods. As a result, there are often great inconsistencies in how different investments are valued and these violations of the law of one price are, in themselves, evidence that markets are inefficient.