Market Efficiency implies stock prices fully reflect all publicly available information instantaneously, thus no investment strategies can systematically earn abnormal returns. Fama (1991) argued that stock prices respond instantaneously and without bias to new value relevant information and that security returns over time are determined only by changes in the market level and individual stock risk. Therefore, no profitable investment opportunities from superior analysis, which implies no one can consistently outperform the market. The precondition for this 'strong' definition is that information and trading cost always equal to zero. This is clearly not true in practice.
Stock market anomalies are violations of Market Efficiency Hypothesis in which consistently abnormal returns can be earned by on some investment strategies that are constructed based on potential market inefficiencies.
The Joint Hypothesis problem refers to the fact that testing for market efficiency is problematic, or even impossible. Any attempts to test for market (in)efficiency must involve an equilibrium asset pricing models. It is not possible to measure 'abnormal' returns without expected returns predicted by pricing models. Therefore, any anomalous evidences on market returns may reflect market inefficiency, bad asset pricing model or both.
In sum, Joint Hypothesis problem implies that Market Efficiency per se is not testable.