Prediction markets are a mechanism for generating an accurate, well calibrated probability about the outcome of an event. For example, if you want to know whether a project will finish on time or if competitor A will merge with competitor B, you can use a prediction market to assess the likelihood that each of those events will occur. Ultimately, the output of the market will be a probability: there's a 28% chance that Project X will finish on time.
Prediction markets produce these probabilities by mimicking a stock market. Within the market, participants (also known as traders) can buy and sell shares of stocks that represent the possible outcomes of the event. For example, if the question is "Will Project X finish on time?," then "Yes" and "No" would be stocks in that market. If I believe that Project X will finish on time, then I buy shares of "Yes."
Each stock has a current price between $0 and $100 per share, which corresponds to the probability that it will be the correct answer. So if the price of "Yes" is $28/share, that means the market thinks there is a 28% chance that "Yes" will be the correct answer. Similar to a real stock market, the price of that stock increases as demand increases. So if many traders believe the project will finish on time, they will buy shares of "Yes," causing its price (and therefore probability) to increase.
Once the "correctness" of an answer becomes known, the market is resolved. So if Project X finishes on time, then "Yes" would be resolved as the correct answer. At that time, traders receive $100 for any shares of "Yes" that they own. So if I purchased 10 shares of "Yes" at $28/share, then I'm paid $1000, netting me a profit of $720. This can be real money in some cases, or it can be a virtual currency in a "play" money prediction market, like Alphacast.