If I have a single share, lets say in Google which is currently worth 605.79. Lets say they pay out the dividends quarterly. What does that mean and how much would I get quarterly?
Google is a poor example since it doesn't pay a dividend (and doesn't expect to), so let's use another example with easy numbers.
Company X has a stock price of $100, and it pays a quarterly dividend (many companies do). Let's assume X pays a dividend of $4. Dividends are always quoted in annual terms, as is dividend yield. When a company says that they pay "quarterly dividends," it means that the company pays dividends every quarter, or every 3 months.
BUT, if a company has a $4 dividend, you will not receive $4 every quarter per share. You will receive $4/4 = $1 per share, every quarter. So over the course of a fiscal year, or 4 quarters, you'll get $1 + $1 + $1 + $1 = $4 per share, which is the annual dividend.
The dividend yield = annual dividend/stock price. So in this case, company X's div. yield will be $4/$100 * 100 = 4%. It's important to note that this is the annual yield. To get the quarterly yield, you must divide by 4. It's also important to note that the yield fluctuates based on stock price, but the dividend payment stays constant unless the company states an announcement.
For a real world example, consider Intel Corp. (TICKER: INTC) http://finance.yahoo.com/q?s=INTC
The share price is currently $22.05, and the dividend is $0.84. This makes the annual yield = $0.84/$22.05 * 100 = 3.80%. Intel pays a quarterly dividend, so you can expect to receive $0.21 every quarter for every share of Intel that you own.
Hope that clears it up!