There are indices for every sort of market, but retail investors are probably most familiar with those related to stock markets.
Indices exist for every sort of market from bonds to more esoteric assets but the major of investors are probably most familiar with stockmarket indices. Indices simply reflect the overall value of the specific group of stocks within the index. They are re-calculated on a regular basis throughout the day.
In the UK, the most popular (or "benchmark" index) is the Financial Times Stock Exchange 100 Share Index (known as the FTSE 100 or the Footsie for more on that, see here).
In the US, the nearest equivalent benchmark index is the Dow Jones Industrial Average, which is calculated by the Dow Jones Company, the publisher of The Wall Street Journal. It focuses on the stocks of the 30 largest and most influential companies in the US. The S&P 500 a more comprehensive US index looks at a much bigger selection of companies.
Germany's benchmark stockmarket index is the Dax 30, and in Japan investors look towards the Topix and Nikkei 225 indices. A popular global stock index, meanwhile, is the MSCI World Index.
See Tim Bennett's video tutorial: What is an index?