Dividends are payments made by a company to its shareholders. When a company earns a profit, that money can be put to two uses: it can either be re-invested in the business (called retained earnings), or it can be paid to the shareholders of the company as a dividend. Paying dividends is not an expense; rather, it is the division of an asset among shareholders. Many companies retain a portion of their earnings and pay the remainder as a dividend. Publicly-traded companies usually pay dividends on a fixed schedule, but may declare a dividend at any time, sometimes called a special dividend to distinguish it from a regular one.
Dividends are usually settled on a cash basis, as a payment from the company to the customer. They can also take the form of shares in the company (either newly-created shares or existing shares bought in the market), and many companies offer dividend reinvestment plans, which automatically use the cash dividend to purchase additional shares for the shareholder.
Since owning a contract for difference is practically identical to owning the actual share, you are also entitled to dividends as per the share holders above. The major difference is that all dividends are cleared in cash, there are no reinvestment schemes or any other sort of substitute available.