An index is primarily used to track the performance of the entirety of, or subset of, a financial market. This is achieved by creating a portfolio of assets that represents the desired segment of the market; this is hypothetical and is used to track the movement in price as a whole across those assets.
An example would be the S&P 500 index, which tracks the performance of 500 of the largest companies in the United States.
You may have also heard of the Consumer Price Index, which tracks average consumer prices across a variety of goods and services and is used to measure inflation.
Indices are often used to benchmark investment funds, providing a useful indicator as to how the fund has performed against the index.
Investors cannot invest directly into an index, however they can invest in funds that seek to track the performance of an index by buying its constituent parts. Index tracking funds, as these are called, are passive funds and generally low in cost as they are not actively managed. You can find open-ended index trackers, or alternatively access this type of strategy via an Exchange Traded Fund ("ETF").
Index trackers will only seek to replicate the performance of an index, so will not generally offer any outperformance.