An index is a tool to measure the movements of a market.
It is the average of the prices of a given number of stocks or bonds or consumer items...
As an example look at the probably most famous and widely used index of them all, the Dow Jones Industrial Average (DJIA). It consists of the combined prices of the 30 biggest US companies traded on the New York Stock Exchange. And it was developed to show you at a glance how the market of the super big US companies is doing.
The decision, which company is added to the index is simply based on the value of the company itself, as derived by the stock price and the number of shares. There are also some more rules to govern the exchange of two companies, above all to avoid having to change the smallest of these companies every other day.
Dow Jones Industrial Average, probably the best known index in the world, covers the 30 biggest companies traded on the New York Stock Exchange
The Swiss Market Index (SMI) covers the 30 biggest stocks listed on the SWX (Swiss Exchange)
All the national bourses are covered with at least one index, from London (FTSE) to Paris (CAC) or Frankfurt (DAX)...
As an Index you might never have heard of: The SWEP covers the electricity market in Switzerland.
Of course there are more than just the these, indexes exist for almost every conceivable thing in the world, as long as there is a price to it. To see a more conclusive list of indexes and the instrument covering them go to the asset classes.