A derivative is a financial instrument whose value is 'derived' from an underlying. This underlying can be the value of a stock (called the spot price) or anything (there are something known as weather derivatives also!). So basically you bet on the unknown.
In the stock market, there are primarily two types of instruments, futures and options. You bet on the upward or downward movement of prices.
With futures, you bet on what the future price of a stock is going to be. The future date is usually the last thursday of the month in the NSE. So we have the September futures, October futures etc. At any point of time there has to be someone taking the opposite stance; if you are going long, then there has to be a counterparty going short.
The idea is same with options, but it is more complicated to explain here. Check out NCFM on derivatives on the NSE site (www.nseindia.com)
The equity shares reflect the current prices, while the derivatives reflect the expected prices in future.